Artikel Matematika kelas 9 kali ini menjelaskan mengenai cara menyusun persamaan kuadrat baru secara lengkap, disertai dengan contoh soal dan pembahasannya.
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Di artikel sebelumnya, kita sudah belajar cara mencari akar-akar dari persamaan kuadrat. Masih ingat nggak dengan bentuk umum persamaan kuadrat? Yup! Bentuk umum persamaan kuadrat adalah
ax2 + bx + c = 0
dengan a, b, c merupakan bilangan real dan a ≠ 0
Nah, kali ini kebalikannya, nih. Kita akan belajar cara menyusun persamaan kuadrat dari akar-akar yang diketahui. Wah, gimana tuh caranya? Oke, daripada penasaran, yuk simak artikel berikut ini!
Ada dua metode untuk menyusun persamaan kuadrat. Metode yang pertama, jika diketahui akar-akar persamaan kuadratnya. Lalu, metode yang kedua, jika diketahui jumlah dan hasil kali dari akar-akar persamaan kuadratnya. Nanti pas ngerjain soal, kamu pilih deh pakai metode yang mana, menyesuaikan dengan yang diketahui di soal.
1. Cara Menyusun Persamaan Kuadrat Jika Diketahui Akar-akarnya
Misalnya, diketahui akar-akar persamaan kuadrat adalah x1 dan x2. Untuk mendapatkan persamaan kuadratnya, kamu bisa substitusi akar-akar tersebut ke persamaan berikut:
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Kenapa sih harus disubstitusi ke persamaan itu?
Kamu masih ingat nggak, kalau ingin mendapatkan akar-akar dari suatu persamaan kuadrat, salah satu caranya adalah dengan memfaktorkan persamaan kuadrat tersebut.
Nah, bentuk persamaan (x – x1)(x – x2) = 0 adalah hasil dari pemfaktoran persamaan kuadrat. Kalau kita lakukan sedikit operasi aljabar, kita kali silang persamaan itu, maka akan didapat suatu persamaan kuadrat.
Oke, supaya lebih paham, perhatikan contoh soal di bawah ini, yuk!
Contoh soal 1
Tentukan persamaan kuadrat yang akar-akarnya adalah 3 dan -7.
Penyelesaian:
Diketahui akar-akar persamaan kuadrat adalah 3 dan -7. Berarti, kamu bisa tulis x1 = 3 dan x2 = -7. Kemudian, kedua akar tersebut bisa kamu substitusikan ke persamaan (x – x1)(x – x2) = 0, sehingga penyelesaiannya menjadi sebagai berikut:
(x – 3)(x – (-7)) = 0
(x – 3)(x + 7) = 0
x2 + 7x – 3x – 21 = 0
x2 + 4x – 21 = 0
Jadiii, persamaan kuadrat yang akar-akarnya 3 dan -7 adalah x2 + 4x – 21 = 0.
Gimana gengs, mudah bukan caranya? Cukup dengan mensubstitusi nilai akar-akarnya dan sedikit melakukan operasi aljabar, kamu sudah bisa mendapatkan persamaan kuadratnya. Yuk, kita lanjut ke metode kedua, ya!
2. Cara Menyusun Persamaan Kuadrat Jika Diketahui Jumlah dan Hasil Kali Akar-akarnya
Misalkan, akar-akar suatu persamaan kuadrat adalah x1 dan x2. Jika yang diketahui pada soal adalah jumlah dan hasil kali akar-akarnya, maka untuk mendapatkan persamaan kuadratnya, kamu bisa gunakan rumus berikut ini:
Nah, sebenarnya, bentuk persamaan x2 – (x1 + x2)x + (x1 . x2) = 0 merupakan hasil kali silang dari persamaan (x – x1)(x – x2) = 0, yang kita gunakan untuk mencari persamaan kuadrat di metode sebelumnya. Penjabarannya, bisa kamu lihat pada gambar di bawah ini, nih.
Terus, kenapa sih bisa dapat x1 + x2= -b/a dan x1 . x2 = c/a? Berawal dari persamaan x2 – (x1 + x2)x + (x1 . x2) = 0, kemudian masing-masing ruas dikalikan dengan konstanta a, sehingga persamaan tersebut menjadi sebagai berikut:
ax2 – a(x1 + x2)x + a(x1 . x2) = 0
Setelah itu, disamain deh dengan bentuk umum persamaan kuadrat, sehingga diperoleh:
Dari penjabaran itu lah rumus hasil jumlah dan kali akar-akar persamaan kuadrat berasal. Gimana, sudah paham ya dengan konsep rumusnya? Oke, sekarang, kita perhatikan contoh soal dibawah ini, ya!
Contoh soal 2
Tentukan persaman kuadrat yang akar-akarnya adalah α dan β, serta jumlah dan hasil kali akar-akarnya adalah -1 dan -20.
Penyelesaian:
Diketahui akar-akarnya adalah x1 dan x2. Kemudian, hasil jumlah akar-akarnya adalah -1, berarti x1 + x2 = -1. Lalu, hasil kali akar-akarnya adalah -20, berarti x1 . x2 = – 20. Nah, kamu bisa langsung substitusi hasil jumlah dan kali akar-akar yang sudah diketahui ke persamaan:
x2 – (x1 + x2)x + (x1 . x2) = 0
Sehingga persamaannya menjadi seperti berikut:
x2 – (-1)x + (-20) = 0
x2 + x – 20 = 0
Jadi, diperoleh persamaan kuadratnya adalah x2 + x -20 = 0.
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Contoh soal 3
Susunlah persamaan kuadrat yang akar-akarnya 3 kali akar-akar persamaan persamaan kuadrat 2x2 + 5x – 3 = 0.
Penyelesaian:
Karena akar persamaan kuadrat yang baru adalah transformasi akar persamaan kuadrat yang lama, kita bisa gunakan metode substitusi.
Apa sih maksudnya transformasi? Maksudnya, dua-duanya berubahnya sama gitu. Di sini, kedua akarnya sama-sama 3 kali akar-akar yang lama.
Biar nggak bingung, kita pakai variabel p untuk persamaan kuadrat yang baru.
Nah, jadinya p = 3x atau kalau kita mau x dalam p, jadinya x = 1/3 p.
Langsung aja kita substitusiin ya,
2x2 + 5x – 3 = 0
2(1/3p)2 + 5(1/3p) – 3 = 0
2/9p2 + 5/3p – 3 = 0
kedua ruas kita kalikan dengan 9
2p2 + 15p – 27 = 0
Sehingga, persamaan kuadrat baru yang akar-akarnya 3 kali persamaan kuadrat 2x2 + 5x – 3 = 0 adalah 2p2 + 15p – 27 = 0.
Kalau mau ditulis lagi dalam x juga nggak papa. Jadinya, 2x2 + 15x – 27 = 0.
Contoh soal 4
Diketahui akar-akar persamaan kuadrat x2 + qx + r = 0 adalah x1 dan x2, dimana x1 < x2. Tentukan persamaan kuadrat dengan akar x1 + 2 dan x2 – 2.
Penyelesaian:
Nah, kalau soalnya kayak gini, nggak bisa pake metode substitusi tadi. Soalnya, x1 dan x2 berubahnya beda. Ada yang ditambah 2, ada yang dikurangi 2.
Terus, gimana, dong? Tenang.
Akar-akar persamaan kuadrat x2 + 3x -10 = 0 adalah x1 dan x2. Kita langsung faktorin aja persamaan kuadratnya, ya. Jadinya,
x2 + 3x -10 = 0
(x-2)(x+5) = 0
Sehingga, diperoleh akar-akarnya, yaitu x = -5 atau x = 2.
Nah, di soal diketahui kalau x1 < x2. Akar yang lebih kecil yang mana? -5 kan ya. Jadi, x1 = -5 dan x2 = 2.
Untuk mencari persamaan kuadrat yang barunya, kita bisa gunakan rumus (x – x1)(x – x2) = 0. Karena diketahui di soal kalau akar-akarnya x1 + 2 dan x2 – 2, berarti:
[x – (x1 + 2)][x – (x2 – 2)]=0
Kita substitusi nilai x1 dan x2 yang kita dapatkan barusan, sehingga:
[x – (-5 + 2)][x – (2 – 2)]=0
(x-(-3))(x-0) = 0
(x+3)(x) = 0
kita kali silang
x2 + 3x = 0
Jadi, persamaan kuadrat dengan akar x1 + 2 dan x2 – 2 adalah x2 + 3x = 0.
Oke, contoh soalnya sudah ada empat, nih. Bisa dong sekarang kalau diminta menyusun persamaan kuadrat. Huehehe…
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Gengs, sadar nggak sih, salah satu kunci agar pandai dalam matematika itu adalah banyak mengerjakan latihan soal. Dengan begitu, logika berpikir kamu akan semakin terasah, rumus-rumus yang sering digunakan pun akan melekat di otak kamu dengan sendirinya. Selain itu, kamu juga bisa bertemu dengan berbagai macam variasi soal. Jadi, pemahaman materi kamu akan semakin dalam.
Nah, kamu bisa lho cobain latihan berbagai macam soal di ruangbelajar. Di sana latihan soalnya lengkap dan ada pembahasannya juga. So, tunggu apa lagi? Buruan gabung sekarang juga!
Referensi:
Wagiyo, A. Mulyono, S. and Susanto, (2008) Pegangan Belajar Matematika 3. Jakarta: Pusat Perbukuan, Departemen Pendidikan Nasional.
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<h1 class="content-title">
Cara Menyusun Persamaan Kuadrat dan Contohnya | Matematika Kelas 9 </h1>
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<p class="content-author"><a href="https://www.ruangguru.com/blog/author/ulfatun-kasanah" class="">Ulfatun Kasanah</a> </p>
<p class="content-info"><span>February 3, 2025 • </span> <span>6 minutes read</span></p>
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<p><img decoding="async" class="aligncenter size-medium" src="https://cdn-web-2.ruangguru.com/landing-pages/assets/7f899646-87fd-491e-b98b-0c701c28dcf7.png" alt="cara mencari persamaan kuadrat" width="820" /></p>
<blockquote>
<p style="font-size: 16px; text-align: center;"><em><strong><a href="https://www.ruangguru.com/blog/c/matematika/matematika-smp-kelas-9">Artikel Matematika kelas 9</a></strong> kali ini menjelaskan mengenai cara menyusun persamaan kuadrat baru secara lengkap, disertai dengan contoh soal dan pembahasannya.</em></p>
</blockquote>
<p style="text-align: center;"><em>—</em></p>
<p> </p>
<p style="text-align: justify;">Di artikel sebelumnya, kita sudah belajar <a style="font-weight: bold;" href="/blog/menyelesaikan-persamaan-kuadrat" rel="noopener">cara mencari akar-akar dari persamaan kuadrat</a>. Masih ingat <em>nggak</em> dengan bentuk umum persamaan kuadrat? <em>Yup!</em> Bentuk umum persamaan kuadrat adalah</p>
<p style="text-align: center;"><span style="font-size: 14pt;">ax<sup>2</sup> + bx + c = 0</span></p>
<p style="text-align: center;">dengan a, b, c merupakan bilangan real dan a ≠ 0</p>
<p style="text-align: justify;"><em>Nah,</em> kali ini kebalikannya, <em>nih</em>. Kita akan belajar cara menyusun persamaan kuadrat dari akar-akar yang diketahui. <em>Wah, </em>gimana <em>tuh</em> caranya? Oke, daripada penasaran, <em>yuk</em> simak artikel berikut ini!</p>
<p style="text-align: justify;">Ada dua metode untuk menyusun persamaan kuadrat. Metode yang pertama, jika diketahui akar-akar persamaan kuadratnya. Lalu, metode yang kedua, jika diketahui jumlah dan hasil kali dari akar-akar persamaan kuadratnya. Nanti <em>pas</em> <em>ngerjain</em> soal, kamu pilih <em>deh </em>pakai metode yang mana, menyesuaikan dengan yang diketahui di soal.</p>
<p> </p>
<h2 style="text-align: justify;"><span style="font-size: 18pt;">1. Cara Menyusun Persamaan Kuadrat Jika Diketahui Akar-akarnya</span></h2>
<p style="text-align: justify;">Misalnya, diketahui akar-akar persamaan kuadrat adalah x<sub>1</sub> dan x<sub>2</sub>. Untuk mendapatkan persamaan kuadratnya, kamu bisa substitusi akar-akar tersebut ke persamaan berikut:</p>
<p><img decoding="async" style="width: 600px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Google%20Drive%20Integration/Cara%20Menyusun%20Persamaan%20Kuadrat%20%20%7C%20Matematika%20%20Kelas%209-3.jpeg" alt="rumus menyusun persamaan kuadrat jika diketahui akar" width="600" /></p>
<p><strong>Baca Juga: <a href="/blog/rumus-luas-dan-volume-kerucut" rel="noopener">Cara Menghitung Luas dan Volume Kerucut</a></strong></p>
<p> </p>
<p style="text-align: justify;">Kenapa <em>sih </em>harus disubstitusi ke persamaan itu?</p>
<p style="text-align: justify;">Kamu masih ingat <em>nggak</em>, kalau ingin mendapatkan akar-akar dari suatu persamaan kuadrat, salah satu caranya adalah dengan memfaktorkan persamaan kuadrat tersebut.</p>
<p style="text-align: justify;"><em>Nah</em>, bentuk persamaan (x – x<sub>1</sub>)(x – x<sub>2</sub>) = 0 adalah hasil dari pemfaktoran persamaan kuadrat. Kalau kita lakukan sedikit operasi aljabar, kita kali silang persamaan itu, maka akan didapat suatu persamaan kuadrat.</p>
<p style="text-align: justify;">Oke, supaya lebih paham, perhatikan contoh soal di bawah ini, <em>yuk</em>!</p>
<p> </p>
<p><strong>Contoh soal 1</strong></p>
<p style="text-align: justify;">Tentukan persamaan kuadrat yang akar-akarnya adalah 3 dan -7.</p>
<p><strong>Penyelesaian:</strong></p>
<p style="text-align: justify;">Diketahui akar-akar persamaan kuadrat adalah 3 dan -7. Berarti, kamu bisa tulis x<sub>1</sub> = 3 dan x<sub>2</sub> = -7. Kemudian, kedua akar tersebut bisa kamu substitusikan ke persamaan (x – x<sub>1</sub>)(x – x<sub>2</sub>) = 0, sehingga penyelesaiannya menjadi sebagai berikut:</p>
<p style="text-align: center;">(x – 3)(x – (-7)) = 0</p>
<p style="text-align: center;">(x – 3)(x + 7) = 0</p>
<p style="text-align: center;">x<sup>2</sup> + 7x – 3x – 21 = 0</p>
<p style="text-align: center;">x<sup>2</sup> + 4x – 21 = 0</p>
<p style="text-align: justify;"><em>Jadiii, </em>persamaan kuadrat yang akar-akarnya 3 dan -7 adalah x<sup>2</sup> + 4x – 21 = 0.</p>
<p style="text-align: justify;">Gimana <em>gengs, </em>mudah bukan caranya? Cukup dengan mensubstitusi nilai akar-akarnya dan sedikit melakukan operasi aljabar, kamu sudah bisa mendapatkan persamaan kuadratnya. <em>Yuk</em>, kita lanjut ke metode kedua, ya!</p>
<p> </p>
<h2 style="text-align: justify;"><span style="font-size: 18pt;">2. Cara Menyusun Persamaan Kuadrat Jika Diketahui Jumlah dan Hasil Kali Akar-akarnya</span></h2>
<p style="text-align: justify;">Misalkan, akar-akar suatu persamaan kuadrat adalah x<sub>1</sub> dan x<sub>2</sub>. Jika yang diketahui pada soal adalah jumlah dan hasil kali akar-akarnya, maka untuk mendapatkan persamaan kuadratnya, kamu bisa gunakan rumus berikut ini:</p>
<p><img decoding="async" style="width: 600px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Google%20Drive%20Integration/Cara%20Menyusun%20Persamaan%20Kuadrat%20%20%7C%20Matematika%20%20Kelas%209-2.jpeg" alt="rumus menyusun persamaan kuadrat" width="600" /></p>
<p> </p>
<p style="text-align: justify;"><em>Nah</em>, sebenarnya, bentuk persamaan x<sup>2</sup> – (x<sub>1</sub> + x<sub>2</sub>)x + (x<sub>1</sub> . x<sub>2</sub>) = 0 merupakan hasil kali silang dari persamaan (x – x<sub>1</sub>)(x – x<sub>2</sub>) = 0, yang kita gunakan untuk mencari persamaan kuadrat di metode sebelumnya. Penjabarannya, bisa kamu lihat pada gambar di bawah ini, <em>nih</em>.</p>
<p><img decoding="async" style="width: 300px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Google%20Drive%20Integration/Cara%20Menyusun%20Persamaan%20Kuadrat%20%20%7C%20Matematika%20%20Kelas%209.jpeg" alt="persamaan kuadrat" width="300" /></p>
<p style="text-align: justify;">Terus, kenapa <em>sih</em> bisa dapat x<sub>1</sub> + x<sub>2</sub>= -b/a dan x<sub>1</sub> . x<sub>2</sub> = c/a? Berawal dari persamaan x<sup>2</sup> – (x<sub>1</sub> + x<sub>2</sub>)x + (x<sub>1</sub> . x<sub>2</sub>) = 0, kemudian masing-masing ruas dikalikan dengan konstanta a, sehingga persamaan tersebut menjadi sebagai berikut:</p>
<p style="text-align: center;"><span style="font-size: 14pt;">ax<sup>2</sup> – a(x<sub>1</sub> + x<sub>2</sub>)x + a(x<sub>1</sub> . x<sub>2</sub>) = 0</span></p>
<p style="text-align: justify;">Setelah itu, disamain <em>deh</em> dengan bentuk umum persamaan kuadrat, sehingga diperoleh:</p>
<p><img decoding="async" style="width: 600px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/menyusun%20persamaan%20kuadrat%201.jpg" alt="cara menyusun persamaan kuadrat" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Dari penjabaran itu <em>lah</em> rumus hasil jumlah dan kali akar-akar persamaan kuadrat berasal. Gimana, sudah paham ya dengan konsep rumusnya? Oke, sekarang, kita perhatikan contoh soal dibawah ini, ya!</p>
<p> </p>
<p><strong>Contoh soal 2</strong></p>
<p style="text-align: justify;">Tentukan persaman kuadrat yang akar-akarnya adalah α dan β, serta jumlah dan hasil kali akar-akarnya adalah -1 dan -20.</p>
<p><strong>Penyelesaian:</strong></p>
<p style="text-align: justify;">Diketahui akar-akarnya adalah x<sub>1</sub> dan x<sub>2</sub>. Kemudian, hasil jumlah akar-akarnya adalah -1, berarti x<sub>1</sub> + x<sub>2</sub> = -1. Lalu, hasil kali akar-akarnya adalah -20, berarti x<sub>1</sub> . x<sub>2</sub> = – 20. <em>Nah</em>, kamu bisa langsung substitusi hasil jumlah dan kali akar-akar yang sudah diketahui ke persamaan:</p>
<p style="text-align: center;">x<sup>2</sup> – (x<sub>1</sub> + x<sub>2</sub>)x + (x<sub>1</sub> . x<sub>2</sub>) = 0</p>
<p style="text-align: justify;">Sehingga persamaannya menjadi seperti berikut:</p>
<p style="text-align: center;">x<sup>2</sup> – (-1)x + (-20) = 0</p>
<p style="text-align: center;">x<sup>2</sup> + x – 20 = 0</p>
<p style="text-align: justify;">Jadi, diperoleh persamaan kuadratnya adalah x<sup>2</sup> + x -20 = 0.</p>
<p style="text-align: justify; font-weight: bold;">Baca Juga: <a href="/blog/menghitung-luas-dan-volume-bola" rel="noopener">Cara Menghitung Luas dan Volume Bola</a></p>
<p> </p>
<p><strong>Contoh soal 3</strong></p>
<p style="text-align: justify;">Susunlah persamaan kuadrat yang akar-akarnya 3 kali akar-akar persamaan persamaan kuadrat 2x<sup>2</sup> + 5x – 3 = 0.</p>
<p><strong>Penyelesaian:</strong></p>
<p style="text-align: justify;">Karena akar persamaan kuadrat yang baru adalah <a style="font-weight: bold;" href="/blog/pengertian-dan-jenis-jenis-transformasi-geometri" rel="noopener">transformasi</a> akar persamaan kuadrat yang lama, kita bisa gunakan metode substitusi.</p>
<p style="text-align: justify;">Apa <em>sih</em> maksudnya transformasi? Maksudnya, dua-duanya berubahnya sama gitu. Di sini, kedua akarnya sama-sama 3 kali akar-akar yang lama.</p>
<p style="text-align: justify;">Biar <em>nggak</em> bingung, kita pakai variabel <em>p</em> untuk persamaan kuadrat yang baru.</p>
<p style="text-align: justify;"><em>Nah</em>, jadinya p = 3x atau kalau kita mau x dalam p, jadinya x = 1/3 p.</p>
<p style="text-align: justify;">Langsung <em>aja</em> kita substitusiin ya,</p>
<p style="text-align: center;">2x<sup>2</sup> + 5x – 3 = 0</p>
<p style="text-align: center;">2(1/3p)<sup>2</sup> + 5(1/3p) – 3 = 0</p>
<p style="text-align: center;">2/9p<sup>2</sup> + 5/3p – 3 = 0</p>
<p style="text-align: center;">kedua ruas kita kalikan dengan 9</p>
<p style="text-align: center;">2p<sup>2</sup> + 15p – 27 = 0</p>
<p style="text-align: justify;">Sehingga, persamaan kuadrat baru yang akar-akarnya 3 kali persamaan kuadrat 2x<sup>2</sup> + 5x – 3 = 0 adalah 2p<sup>2</sup> + 15p – 27 = 0.</p>
<p style="text-align: justify;">Kalau mau ditulis lagi dalam x juga <em>nggak</em> papa. Jadinya, 2x<sup>2</sup> + 15x – 27 = 0.</p>
<p> </p>
<p><strong>Contoh soal 4 </strong></p>
<p style="text-align: justify;">Diketahui akar-akar persamaan kuadrat x<sup>2</sup> + qx + r = 0 adalah x<sub>1</sub> dan x<sub>2</sub>, dimana x<sub>1</sub> < x<sub>2</sub>. Tentukan persamaan kuadrat dengan akar x<sub>1</sub> + 2 dan x<sub>2 </sub>– 2.</p>
<p><strong>Penyelesaian:</strong></p>
<p style="text-align: justify;"><em>Nah</em>, kalau soalnya kayak gini, <em>nggak</em> bisa <em>pake</em> metode substitusi tadi. Soalnya, x<sub>1</sub> dan x<sub>2</sub> berubahnya beda. Ada yang ditambah 2, ada yang dikurangi 2.</p>
<p style="text-align: justify;">Terus, gimana, <em>dong</em>? Tenang.</p>
<p>Akar-akar persamaan kuadrat x<sup>2</sup> + 3x -10 = 0 adalah x<sub>1</sub> dan x<sub>2</sub>. Kita langsung faktorin <em>aja</em> persamaan kuadratnya, ya. Jadinya,</p>
<p style="text-align: center;">x<sup>2</sup> + 3x -10 = 0</p>
<p style="text-align: center;">(x-2)(x+5) = 0</p>
<p style="text-align: justify;">Sehingga, diperoleh akar-akarnya, yaitu x = -5 atau x = 2.</p>
<p style="text-align: justify;"><em>Nah</em>, di soal diketahui kalau x<sub>1</sub> < x<sub>2</sub>. Akar yang lebih kecil yang mana? -5 <em>kan</em> ya. Jadi, x<sub>1</sub> = -5 dan x<sub>2</sub> = 2.</p>
<p style="text-align: justify;">Untuk mencari persamaan kuadrat yang barunya, kita bisa gunakan rumus (x – x<sub>1</sub>)(x – x<sub>2</sub>) = 0. Karena diketahui di soal kalau akar-akarnya x<sub>1</sub> + 2 dan x<sub>2 </sub>– 2, berarti:</p>
<p style="text-align: center;">[x – (x<sub>1</sub> + 2)][x – (x<sub>2</sub> – 2)]=0</p>
<p style="text-align: justify;">Kita substitusi nilai x<sub>1 </sub>dan x<sub>2</sub> yang kita dapatkan barusan, sehingga:</p>
<p style="text-align: center;">[x – (-5 + 2)][x – (2 – 2)]=0</p>
<p style="text-align: center;">(x-(-3))(x-0) = 0</p>
<p style="text-align: center;">(x+3)(x) = 0</p>
<p style="text-align: center;">kita kali silang</p>
<p style="text-align: center;">x<sup>2 </sup>+ 3x = 0</p>
<p style="text-align: justify;">Jadi, persamaan kuadrat dengan akar x<sub>1</sub> + 2 dan x<sub>2 </sub>– 2 adalah x<sup>2 </sup>+ 3x = 0.</p>
<p style="text-align: justify;">Oke, contoh soalnya sudah ada empat, <em>nih. </em>Bisa <em>dong</em> sekarang kalau diminta menyusun persamaan kuadrat. <em>Huehehe</em>…</p>
<p style="text-align: center;">—</p>
<p style="text-align: justify;"><em>Gengs</em>, sadar <em>nggak sih</em>, salah satu kunci agar pandai dalam matematika itu adalah banyak mengerjakan latihan soal. Dengan begitu, logika berpikir kamu akan semakin terasah, rumus-rumus yang sering digunakan pun akan melekat di otak kamu dengan sendirinya. Selain itu, kamu juga bisa bertemu dengan berbagai macam variasi soal. Jadi, pemahaman materi kamu akan semakin dalam.</p>
<p style="text-align: justify;"><em>Nah</em>, kamu bisa <em>lho</em> cobain latihan berbagai macam soal di <strong><a href="http://ruangguru.com/belajar" target="_blank" rel="noopener">ruangbelajar</a></strong>. Di sana latihan soalnya lengkap dan ada pembahasannya juga. <em>So</em>, tunggu apa lagi? Buruan gabung sekarang juga!</p>
<p style="text-align: justify;"><a class="rg-cta" style="text-align: center;" href="https://www.ruangguru.com/download-for-desktop" target="_blank" rel="noopener"><img fetchpriority="high" decoding="async" class="aligncenter" src="https://cdn-web.ruangguru.com/landing-pages/assets/cta/f51e0d89-905d-477d-b7f8-546567f121d9.jpeg" alt="IDN CTA Blog ruangbelajar for desktop Ruangguru" width="800" height="195" /></a></p>
<p style="text-align: justify;"><strong>Referensi:</strong></p>
<p style="text-align: left;">Wagiyo, A. Mulyono, S. and Susanto, (2008) <em>Pegangan Belajar Matematika 3. </em>Jakarta: Pusat Perbukuan, Departemen Pendidikan Nasional.</p>
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<div class="d-lg-none text-center">
<nav class="footer-menu social">
<p class="title">Ikuti Kami</p>
<a target="_blank" href="https://www.instagram.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/instagram.png" loading="lazy" alt="Logo Instagram Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.facebook.com/ruanggurucom/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/facebook.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/facebook.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/facebook.png" loading="lazy" alt="Logo Facebook Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://twitter.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/twitter.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/twitter.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/twitter.png" loading=" lazy" alt="Logo Twitter Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.youtube.com/ruangguru">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/youtube.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/youtube.png" loading=" lazy" alt="Logo Youtube Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.linkedin.com/company/ruangguru-com">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/linkedin.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/linkedin.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/linkedin.png" loading=" lazy" alt="Logo LinkedIn Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.tiktok.com/@ruangguru">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/tiktok.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/tiktok.png" loading=" lazy" alt="Logo TikTok Footer" width="32px" height="32px" />
</picture>
</a>
</nav>
</div>
<p class="footer-copyright">
© 2026 All Rights Reserved PT. Ruang Raya Indonesia
</p>
</div>
<style>
.bantuan-container {
position: relative;
}
.bantuan-element {
position: fixed;
z-index: 999;
display: flex;
align-items: flex-end;
justify-content: flex-end;
cursor: pointer;
transition: all 0.6s ease;
}
.bantuan-top {
bottom: 11rem;
right: 2rem;
}
.bantuan-bottom {
bottom: 2rem;
right: 2rem;
}
.bantuan-top-2 {
bottom: 15rem;
right: 2rem;
}
.bantuan-top-3 {
bottom: 19rem;
right: 2rem;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 2rem;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 2rem;
}
@media (min-width: 768px) {
.bantuan-element img {
width: auto;
transition: all 0.4s ease;
}
.bantuan-top img {
width: 140px;
}
.bantuan-bottom img {
width: 140px;
}
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 140px;
}
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 140px;
}
.bantuan-element.clicked {
bottom: 9rem;
right: 2.5rem;
}
.bantuan-element.clicked img {
width: auto;
height: 64px;
}
}
@media (max-width: 767px) {
.bantuan-top {
bottom: 8rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom {
bottom: 2rem;
right: 1.25rem;
align-items: center;
}
.bantuan-top-2 {
bottom: 12rem;
right: 1rem;
align-items: center;
}
.bantuan-top-3 {
bottom: 16rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 1.25rem;
align-items: center;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 1.25rem;
align-items: center;
}
.bantuan-element img {
width: auto;
}
.bantuan-top img,
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 96px;
}
.bantuan-bottom img,
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 96px;
}
}
.bantuan-element:hover {
transform: scale(1.05);
}
.bantuan-element img {
max-width: 100%;
height: auto;
}
</style>
<div id="bantuan-divs-container" style="position: relative" data-product="ruangguru" data-table="rg_floating_icon"></div>
<script>
class FloatingDivsComponent {
constructor(product = 'ruangguru', table = 'rg_floating_icon') {
this.container = document.getElementById('bantuan-divs-container');
this.floatingElements = [];
this.product = product;
this.table = table;
// Configuration
this.config = {
baseApiUrl: 'https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/rg_banner/views/',
apiKey: 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
maxElements: 2,
zIndexBase: 100,
};
// Build dynamic API URL based on table parameter
this.config.apiUrl = this.buildApiUrl();
this.init();
}
buildApiUrl() {
const baseUrl = this.config.baseApiUrl;
const tableName = this.table || 'rg_floating_icon';
const queryParams =
'offset=0&limit=99999&sort=rankPosition&where=(type,eq,floating)~and(showIn,anyof,blog)~and(hide,is,false)~and(product,eq,' +
this.product +
')&nested%5Bcta%5D%5Bfields%5D=Id,name,ctaId,imageUrl,ctaLink,targetUrl';
return `${baseUrl}${tableName}?${queryParams}`;
}
async init() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to initialize floating divs component:', error);
}
}
async fetchFloatingElements() {
try {
const options = {
method: 'GET',
headers: {
'xc-token': this.config.apiKey || 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
},
};
const urlNoco = this.config.apiUrl;
const response = await fetch(urlNoco, options);
if (!response.ok) {
throw new Error(`HTTP error! status: ${response.status}`);
}
const data = await response.json();
this.floatingElements = this.processApiResponse(data);
} catch (error) {
console.error('Error fetching floating elements:', error);
throw error;
}
}
processApiResponse(data) {
if (!data || !data.list || !Array.isArray(data.list)) {
return [];
}
console.log(data.list);
// Ensure we only get maximum 2 elements
const limitedElements = data.list.slice(0, this.config.maxElements);
return limitedElements;
}
renderFloatingElements() {
if (!this.container || this.floatingElements.length === 0) {
return;
}
// Additional safeguard: ensure we never render more than maxElements
const elementsToRender = this.floatingElements.slice(0, this.config.maxElements);
// Clean up existing elements and their event listeners
this.cleanupExistingElements();
// Clear existing content
this.container.innerHTML = '';
elementsToRender.forEach((element, index) => {
const floatingDiv = this.createFloatingDiv(element, index);
this.container.appendChild(floatingDiv);
});
}
cleanupExistingElements() {
// Clear existing elements (no need for event listener cleanup with CSS approach)
const existingElements = this.container.querySelectorAll('[data-floating-element]');
existingElements.forEach(element => {
element.remove();
});
}
createFloatingDiv(element, index) {
const div = document.createElement('div');
const cta = element.cta[0]; // Use the first CTA
// Add data attribute for cleanup tracking
div.setAttribute('data-floating-element', 'true');
// Apply CSS classes based on rankPosition
const positionClass = this.getPositionClass(element.rankPosition, index);
div.className = `bantuan-element ${positionClass}`;
div.style.zIndex = this.config.zIndexBase + index;
// Create the link and image
const link = document.createElement('a');
link.href = 'https://cta.ruangguru.com/' + cta.ctaId || '#';
link.target = '_blank';
// Add tracking attributes
link.setAttribute('data-tracking-event-type', 'landing_page__core__item_click');
link.setAttribute(
'data-tracking-context',
JSON.stringify({
product_name: this.product,
cta_action: 'redirection',
cta_action_detail: link.href,
item_type: 'floating',
item_name: element.name || 'Floating element',
item_category: 'payment',
item_position: index,
item_serial: null,
section_type: 'floating_item',
section_position: 'body',
section_name: 'floating_item',
})
);
const img = document.createElement('img');
img.src = element.imageUrl + '?h=200&convert=webp';
img.alt = element.name || 'Floating element';
img.loading = 'lazy';
// Add click tracking
link.addEventListener('click', () => {
this.trackClick(element, cta);
});
link.appendChild(img);
div.appendChild(link);
return div;
}
getPositionClass(rankPosition, index) {
// Define positioning classes based on rankPosition
if (rankPosition === 1) {
return 'bantuan-top'; // Top position
} else if (rankPosition === 2) {
return 'bantuan-bottom'; // Bottom position
} else {
// For additional elements, distribute them evenly
return `bantuan-${index % 2 === 0 ? 'top' : 'bottom'}-${Math.floor(index / 2) + 1}`;
}
}
trackClick(element, cta) {
// Implement click tracking logic here
console.log('Floating element clicked:', {
elementId: element.Id,
elementName: cta.name,
ctaId: cta.ctaId,
targetUrl: cta.targetUrl,
});
}
// Public method to manually refresh
async refresh() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to refresh floating elements:', error);
throw error;
}
}
// Public method to get current floating elements
getFloatingElements() {
return [...this.floatingElements];
}
}
// Initialize the component when DOM is ready
document.addEventListener('DOMContentLoaded', () => {
// Get product and table parameters from data attributes
const container = document.getElementById('bantuan-divs-container');
const product = container.getAttribute('data-product') || 'ruangguru';
const table = container.getAttribute('data-table') || 'rg_floating_icon';
new FloatingDivsComponent(product, table);
});
// Export for use in other modules if needed
if (typeof module !== 'undefined' && module.exports) {
module.exports = FloatingDivsComponent;
}
</script> <div id="wp-install-nudge" style="
position: fixed;
bottom: 0rem;
right: 0;
z-index: 999;
cursor: pointer;
"></div>
<script>
(function (w, d, s, o, f, js, fjs) {
w['Sales-Nudge-Widget'] = o; w[o] = w[o] || function () { (w[o].q = w[o].q || []).push(arguments) };
js = d.createElement(s), fjs = d.getElementsByTagName(s)[0];
js.id = o; js.src = f; js.async = 1; fjs.parentNode.insertBefore(js, fjs);
}(window, document, 'script', 'wpInstallNudge', 'https://cdn-web-2.ruangguru.com/sales-nudge/6198a95130/dist/widget.js'));
wpInstallNudge('init', { targetElementId: 'wp-install-nudge', clientTarget: 'ruangguru_blog' });
</script>
<script>
// note: disabled due to auto reload page
// var lastScrollTop = 0;
// $(window).scroll(function(event){
// var st = $(this).scrollTop();
// if (st > lastScrollTop){
// $('#wp-install-nudge').slideDown( "slow", function() {});
// } else {
// $('#wp-install-nudge').slideUp( "slow", function() {});
// }
// lastScrollTop = st;
// });
// $(document).ready(function () {
// $("#wp-install-nudge").hide();
// });
$('#wp-install-nudge').click(function(){
Analytics.track("promotion__install_nudge__click",
{
user_email: "ruangguru@blog.com",
cta_link: widgetData.dataConfig.target_link
}).catch((err) => {
console.log("tracker error", err);
});
});
</script>
</footer>
<div id="__next"></div>
<script type="speculationrules">
{"prefetch":[{"source":"document","where":{"and":[{"href_matches":"\/blog\/*"},{"not":{"href_matches":["\/blog\/wp-*.php","\/blog\/wp-admin\/*","\/blog\/wp-content\/uploads\/*","\/blog\/wp-content\/*","\/blog\/wp-content\/plugins\/*","\/blog\/wp-content\/themes\/ruangguru\/*","\/blog\/*\\?(.+)"]}},{"not":{"selector_matches":"a[rel~=\"nofollow\"]"}},{"not":{"selector_matches":".no-prefetch, .no-prefetch a"}}]},"eagerness":"conservative"}]}
</script>
<script>
async function getCtaDetails(ctaIds) {
const endpoint =
" https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/CTA/views/CTA?offset=0&limit=10&where=(ctaId,in," + ctaIds.toString() + ")";
try {
const response = await fetch(endpoint, {
headers: {
"xc-token": "-hG2nBDJzSj0agSgNtcJx97M4F02y54nUMoY6plJ",
},
});
const data = await response.json();
const ctaList = data?.list || [];
const ctaMap = new Map();
for (const ctaData of ctaList) {
ctaMap.set(ctaData.ctaId, ctaData);
}
return ctaMap;
} catch (error) {
console.error("Error fetching CTA details:", error);
return new Map();
}
}
async function replaceCTA() {
const regex = /{{cta\((‘(.*?)’|“(.*?)”)\)}}/g;
const bodyHTML = document.body.innerHTML;
const matches = Array.from(bodyHTML.matchAll(regex));
const ctaIds = matches.map((match) => {
let ctaId = match[2] ?? match[3];
return ctaId;
});
const ctaMap = await getCtaDetails(ctaIds);
const replacedHTML = bodyHTML.replace(regex, (match, _inner, ctaId1, ctaId2) => {
const ctaId = ctaId1 ?? ctaId2;
const ctaData = ctaMap.get(ctaId);
if (ctaData) {
return `<a href="https://cta.ruangguru.com/${ctaData.ctaId}" target="_${ctaData.targetType}" style="position:relative;display:inline-block;">
<img src="${ctaData.imageUrl}" loading="lazy" alt="${ctaData.altText}" />
</a>`;
} else {
return match;
}
});
document.body.innerHTML = replacedHTML;
}
</script>
<meta name="author" content="Ruangguru Tech Team">
<!-- Google Tag Manager (noscript) -->
<noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PVNLZFT" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>
<!-- Tracker -->
<script type="text/javascript" src="https://cdn-web-2.ruangguru.com/landing-page-web/public/staticpages/global-assets/js/cta.js"></script>
<script src="https://unpkg.com/@dotlottie/player-component@latest/dist/dotlottie-player.mjs" type="module"></script>
</body>
</html>