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Original
2026-01-01
Modified
2026-02-28
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<p>Depending on the structure of the quadratic equation, it can be factored using various methods. These include: </p>
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<p>Depending on the structure of the quadratic equation, it can be factored using various methods. These include: </p>
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<h2><strong>Factoring by splitting the middle term</strong></h2>
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<h2><strong>Factoring by splitting the middle term</strong></h2>
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<p>The middle terms are split such that their<a>sum</a>equals the coefficients of x and their product equals the product of x2 and the<a>constant</a>term.</p>
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<p>The middle terms are split such that their<a>sum</a>equals the coefficients of x and their product equals the product of x2 and the<a>constant</a>term.</p>
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<ul><li>Let's begin with a quadratic equation in the form ax² + bx + c = 0. </li>
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<ul><li>Let's begin with a quadratic equation in the form ax² + bx + c = 0. </li>
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<li>We then find two<a>numbers</a>whose sum is b and whose product is equal to a × c. </li>
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<li>We then find two<a>numbers</a>whose sum is b and whose product is equal to a × c. </li>
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<li>The middle term is rewritten as split terms </li>
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<li>The middle term is rewritten as split terms </li>
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<li>Group the terms into pairs. </li>
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<li>Group the terms into pairs. </li>
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<li>Find the GCF from each group. </li>
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<li>Find the GCF from each group. </li>
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<li>Factor out the common<a>binomial</a>expression.</li>
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<li>Factor out the common<a>binomial</a>expression.</li>
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</ul><p>Example: Factorize 3x² + 11x + 6</p>
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</ul><p>Example: Factorize 3x² + 11x + 6</p>
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<ul><li>To split the middle term, find numbers having a sum of 11 and a product of 3 × 6 = 18. 9 and 2 are the required numbers. </li>
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<ul><li>To split the middle term, find numbers having a sum of 11 and a product of 3 × 6 = 18. 9 and 2 are the required numbers. </li>
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<li>Rewrite the middle term: 3x² + 9x + 2x + 6 </li>
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<li>Rewrite the middle term: 3x² + 9x + 2x + 6 </li>
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<li>Group the terms (3x² + 9x) + (2x + 6) </li>
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<li>Group the terms (3x² + 9x) + (2x + 6) </li>
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<li>Factor 3x(x + 3) + 2(x + 3) </li>
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<li>Factor 3x(x + 3) + 2(x + 3) </li>
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<li>Factor (3x + 2)(x + 3)</li>
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<li>Factor (3x + 2)(x + 3)</li>
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</ul><p>Therefore, 3x² + 11x + 6 = (3x + 2)(x + 3)</p>
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</ul><p>Therefore, 3x² + 11x + 6 = (3x + 2)(x + 3)</p>
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<h2><strong>Factoring using the formula</strong></h2>
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<h2><strong>Factoring using the formula</strong></h2>
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<p>We use this method when x² has the<a>coefficient</a>1 and the quadratic is expressed as (x + a)(x + b).</p>
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<p>We use this method when x² has the<a>coefficient</a>1 and the quadratic is expressed as (x + a)(x + b).</p>
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<p>Steps:</p>
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<p>Steps:</p>
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<p>Write the quadratic in its<a>standard form</a>,<a>i</a>.e., x² + bx + c.</p>
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<p>Write the quadratic in its<a>standard form</a>,<a>i</a>.e., x² + bx + c.</p>
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<p>Find two numbers, whose sum is b and product is c.</p>
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<p>Find two numbers, whose sum is b and product is c.</p>
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<p>Rewrite the quadratic using the factors.</p>
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<p>Rewrite the quadratic using the factors.</p>
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<p>Example: Factorize x² + 7x + 12</p>
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<p>Example: Factorize x² + 7x + 12</p>
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<ul><li>We use the numbers 3 and 4, because, 3 + 4 = 7 and 3 × 4 = 12 </li>
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<ul><li>We use the numbers 3 and 4, because, 3 + 4 = 7 and 3 × 4 = 12 </li>
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<li>Factor: (x + 3)(x + 4)</li>
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<li>Factor: (x + 3)(x + 4)</li>
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</ul><p>Thus, x² + 7x + 12 = (x + 3)(x + 4)</p>
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</ul><p>Thus, x² + 7x + 12 = (x + 3)(x + 4)</p>
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<h2><strong>Factoring using the quadratic formula</strong></h2>
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<h2><strong>Factoring using the quadratic formula</strong></h2>
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<p>Also known as Shridharacharya’s formula, this is a universal method for solving and factorizing quadratic equations.</p>
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<p>Also known as Shridharacharya’s formula, this is a universal method for solving and factorizing quadratic equations.</p>
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<p>The formula is: x = -b b2- 4ac2a </p>
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<p>The formula is: x = -b b2- 4ac2a </p>
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<p>Steps:</p>
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<p>Steps:</p>
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<ul><li>The equation should be written in standard form, that is, ax² + bx + c = 0 </li>
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<ul><li>The equation should be written in standard form, that is, ax² + bx + c = 0 </li>
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<li>Then the quadratic formula is used for finding roots x₁ and x₂. </li>
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<li>Then the quadratic formula is used for finding roots x₁ and x₂. </li>
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<li>This gives us the factored form a(x - x₁)(x - x₂)</li>
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<li>This gives us the factored form a(x - x₁)(x - x₂)</li>
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</ul><p>Example: Factorize 2x² - 5x - 3</p>
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</ul><p>Example: Factorize 2x² - 5x - 3</p>
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<p>Use the quadratic formula: x = - (-5) (-5 )2 - 4(2)(-3)2(2) = 5 25 + 244 = 5 454 = 5 74</p>
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<p>Use the quadratic formula: x = - (-5) (-5 )2 - 4(2)(-3)2(2) = 5 25 + 244 = 5 454 = 5 74</p>
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<p>Roots: x₁ = 3, x₂ = -½</p>
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<p>Roots: x₁ = 3, x₂ = -½</p>
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<p>Rewrite: 2(x - 3)(x + ½)</p>
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<p>Rewrite: 2(x - 3)(x + ½)</p>
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<p>Thus, 2x² - 5x - 3 = 2(x - 3)(x + ½)</p>
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<p>Thus, 2x² - 5x - 3 = 2(x - 3)(x + ½)</p>
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<p>Now we will simplify it further. (x - 3) (x + 12) = x2 + 12x - 3x - 32 = x2 - 52x - 32</p>
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<p>Now we will simplify it further. (x - 3) (x + 12) = x2 + 12x - 3x - 32 = x2 - 52x - 32</p>
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<p>Multiplying everything by 2, we get 2 (x2 - 52x - 32) = 2x2 - 5x - 3</p>
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<p>Multiplying everything by 2, we get 2 (x2 - 52x - 32) = 2x2 - 5x - 3</p>
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<p>So the simplified expression is 2x2 - 5x - 3</p>
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<p>So the simplified expression is 2x2 - 5x - 3</p>
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<h2><strong>Factoring Using Algebraic Identities</strong></h2>
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<h2><strong>Factoring Using Algebraic Identities</strong></h2>
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<p>This method uses well-known algebraic identities to factorize special forms of quadratic expressions quickly.</p>
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<p>This method uses well-known algebraic identities to factorize special forms of quadratic expressions quickly.</p>
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<p>Common identities used:</p>
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<p>Common identities used:</p>
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<ul><li>(a + b)² = a² + 2ab + b² </li>
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<ul><li>(a + b)² = a² + 2ab + b² </li>
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<li>(a - b)² = a² - 2ab + b² </li>
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<li>(a - b)² = a² - 2ab + b² </li>
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<li>a² - b² = (a + b)(a - b)</li>
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<li>a² - b² = (a + b)(a - b)</li>
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</ul><p>Steps:</p>
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</ul><p>Steps:</p>
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<ul><li>Recognize the algebraic identity that can be applied to the given equation. </li>
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<ul><li>Recognize the algebraic identity that can be applied to the given equation. </li>
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<li>Apply the appropriate identity to factorize.</li>
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<li>Apply the appropriate identity to factorize.</li>
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</ul><p>Example: Factorize 9x² - 16</p>
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</ul><p>Example: Factorize 9x² - 16</p>
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<p>This equation is a difference of squares: (3x)² - (4)²</p>
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<p>This equation is a difference of squares: (3x)² - (4)²</p>
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<p>So, we apply the identity: a² - b² = (a + b)(a - b),</p>
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<p>So, we apply the identity: a² - b² = (a + b)(a - b),</p>
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<p>This gives us, 9x² - 16 = (3x + 4)(3x - 4)</p>
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<p>This gives us, 9x² - 16 = (3x + 4)(3x - 4)</p>
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