Quadratic Equations
2026-02-28 10:30 Diff
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Last updated on December 12, 2025

A quadratic equation is a second-degree polynomial written in the form of ax² + bx + c = 0, where a ≠ 0. It is used to solve problems involving area, velocity, and motion paths. For example, it is used to calculate the area of different shapes, the velocity of moving objects, and the trajectories of objects following parabolic paths, such as projectiles. The solutions to a quadratic equation are the points where its parabola-shaped graph crosses the x-axis. These solutions are called roots or zeros.

What is Quadratic Equation

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A quadratic equation is a second-degree polynomial equation written as:
\(ax^2 + bx + c = 0\)

Where:
a, b are coefficients (with a ≠ 0),
x is the variable
c is the constant
And the highest power of x is 2.

The word ‘quadratic’ is derived from the Latin word ‘Quadratus’, which means square. The equation is quadratic because ‘quad’ means the square (power 2). The solutions to a quadratic equation, known as its roots, can be real or complex numbers. The graph of a quadratic equation is known as a parabola.

Examples of quadratic equations in different forms are: (x + 4)(x - 3) = 0, 2x2 + 5x - 3 = 0, 4x(x - 2) = 8, 5x - 5 = 2x2.

What are the Roots of a Quadratic Equation?

The roots of a quadratic equation of the form \(ax^2  + bx + c = 0\) are values of x that equate the equation to zero. Hence, they are also called zeros, and they satisfy the equation. Solving a quadratic equation yields two values of x, which can be real or complex. For example., the equation \(x^2 - 3x - 4 = 0 \) has two roots; x = -1 and x = 4. 

We can verify this by substituting the values of x in the equation:

When x = -1, we get: \((-1)^2 - 3(-1) - 4 = 0\)
Simplifying it: \(1 + 3 - 4 = 0 \)
\(4 - 4 = 0\)

When x = 4, we get: \((4)^2 - 3(4) - 4 = 0 \)
Simplifying it: \(16 - 12 - 4 = 0  \)
\(4 - 4 = 0\)

There are various methods to find the roots of a quadratic equation. One of them is by using the quadratic formula.

What is the Quadratic Formula?

There are a few quadratic equations that cannot be factorized with ease, and here use this formula to find the roots. The two roots in the quadratic formula are written in the form of a single expression. Solving a quadratic expression will usually yield two roots; one with a positive value and one with a negative value.

The quadratic formula for the equation \(ax² + bx + c = 0 \) is: 
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\).

Example: Let’s find the roots of the quadratic equation \(y^2 + 2y - 15 = 0 \)using the quadratic formula.

Here,
a = 1,
b = 2,
c = −15
Using the formula: \(y = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Substituting the values:  \(y = {-2 \pm \sqrt{2^2-4(1)(-15)} \over 2(1)}\)

                                                                        \(= {-2 \pm \sqrt{4+ 60} \over 2}\)

                                       

\(= {-2 \pm \sqrt{64} \over 2}\)

                                                                                 \( = {{-2 ± 8\over2}}\)

So, y = \({{-2 + 8\over2 }}=  {{6\over2}} = 3\)

         and,   y \(= {{-2 - 8\over2}} = {{-10\over2 }}= -5\)

Therefore, the roots of the equation are 3 and -5.
 

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Proof of Quadratic Formula

The quadratic formula, used in solving equations of the form \(ax^2  + bx + c = 0\), is obtained by following the process of completing the square. In this manner, we convert the given equation into the perfect square form, and thus we can isolate the variable and get a general solution that is applicable for any given coefficient.

Example: Examine the following arbitrary quadratic equation: \(ax^2  + bx + c = 0, {\text { a ≠ 0}}\).

We take the following steps to find the equation's roots:

Step 1: \(ax^2 + bx = -c ⇒ x^2 + {{bx\over a}} = {{-c\over a}}\)

By adding a new term (b/2a)2 to both sides, we can now represent the left-hand side as a perfect square:

\(x^2 + {{bx\over a}} + {{({b\over 2a})^2}} = {{-c\over a }}+ {{({b\over 2a})^2}}\)

Now, the left side is a perfect square:

\({(x + {b\over 2x})}^2 = {-c \over a} + {b^2 \over 4a^2} \implies ({x + b \over 2a})^2 = {({b^2 - 4ac \over 4a^2})}\)

Simplifying the equation further, we get:

\({x + {b\over 2a}} = ± \sqrt {b^2 - 4ac \over 2a}\)

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

We were therefore able to isolate x and derive the equation's two roots by completing the squares.
 

What is the Nature of Roots of the Quadratic Equation?

Greek letters alpha (α) and beta (β) are commonly used to represent the roots of a quadratic equation. We can determine the type of roots using the discriminant. Here, we'll learn more about determining the type of roots in a quadratic equation.

It is possible to determine the nature of a quadratic equation's roots without actually determining the equation's roots (α, β). This can be accomplished by calculating the discriminant value, which is a component of the quadratic equation solution formula. The discriminant of a quadratic equation, denoted by the letter ‘D,’ is the value \(b^2 - 4ac\).

It is possible to predict the characteristics of the quadratic equation's roots based on the discriminant value.
Discriminant: \(D = b^2 - 4ac\).

  • If D > 0, then the roots are real and distinct.
  • If D = 0, then the roots are real and equal.
  • When D < 0, the roots are complex.

Now observe the formulas for finding the sum and the product of the roots of the given equation.

Sum and Product of Roots of Quadratic Equation 

The sum and product of roots of a quadratic equation can be found using the coefficient of \(x^2\), the coefficient of x, and the constant term of the equation \(ax^2 + bx + c = 0\).

Compute the sum and product of the roots of the equation from the equation. The sum and product of the roots of the quadratic equation \(ax^2 + bx + c = 0 \) are as follows.

The sum of the roots is \(\alpha + \beta = \frac{-b}{a} = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2} \).
The product of the roots is \(\alpha \beta = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of } x^2} \).

Writing Quadratic Equations Using Roots

Using the roots of a quadratic equation, we can write the quadratic equation. If α and β are the roots, the quadratic equation can be written as:

x2 - (α + β)x + αβ = 0

For example, find the quadratic equation whose roots are 3 and 5

Let α = 3
β = 5

Using the formula for a quadratic equation based on roots: 

x2 - (α + β)x + αβ = 0

x2 - (3 + 5)x + (3)(5) = 0

x2 - 8x + 15 = 0

So, the quadratic equation with roots 3 and 5 is: x2 - 8x + 15 = 0

Methods to Solve Quadratic Equations

You can find the two roots of the quadratic equations using the four methods listed below:

  • Quadratic equation factorization
  • Applying the quadratic formula, which we have already seen in the above topics.
  • Completing the square
  • Graphing the equation

To learn more about the aforementioned techniques, their applications, and their uses, let's take a closer look at them.

How to Solve Quadratic Equations by Factorization

There are several steps involved in factorizing a quadratic equation. We must first split the middle part of the equation into two groups. The split must happen in such a way that their product should be the same as the product of coefficient a and constant c in the equation \(ax^2 + bx + c = 0\).

Additionally, we can extract the common terms from the available terms to ultimately derive the necessary factors in the manner described below:

  • \( x^2 + (a + b)x + ab = 0 \)
  • \((x^2 + ax) + (bx + ab) = 0 \)
  • \(x(x + a) + b(x + a) = 0 \)
  • \((x + a)(x + b) = 0 \)


To better understand the factorization process, consider this example.

  • \(x^2 + 5x + 6\)
  • \(x^2 + 2x + 3x + 6 = 0  \)
  • \(x(x + 2) + 3(x + 2) = 0 \)
  • \((x + 2) (x + 3) = 0\)


As a result, the quadratic equation's two obtained factors are \((x + 2)\) and \((x + 3)\). Simply set each factor to zero and solve for x to determine its roots. That is, \(x = -2 \) and \(x = -3\) because \(x + 2 = 0\) and \(x + 3 = 0\). Therefore, the roots of \(x^2 + 5x + 6 = 0\) are \(x = -2 \) and \(x = -3\).

Additionally, there is another crucial approach to solving a quadratic equation. Finding the roots of a quadratic equation can also be accomplished by using the method of completing the square.

What are the Methods of Completing the Square

Completing the square in a quadratic equation involves simplifying and algebraically squaring the equation to find the necessary roots. Examine the quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\). We simplify this equation as follows to find its roots:

\(ax^2 + bx + c = 0\)
\(ax^2 + bx = -c\)

\( {{{{x^2+ {{bx \over a}}}} = {{-c \over a}}}}\)

By adding a new term \({{({b \over 2a})}}^2\) to both sides, we can now represent the left-hand side as a perfect square:

\({{x^2 + {{bx \over a}} + {{({b \over 2a})^2}}}} = {{{{-c\over a }}+ {{({b\over 2a})^2}}}}\) 
\({({x + {b\over 2a}})}^2 = {-c \over a} + {b^2 \over 4a^2}\)
\({({{x + b} \over 2a})}^2 = {{{({b^2 - 4ac})} \over 4a^2}}\)
\({x + {b \over 2a}} = {{± \sqrt {b^2 - 4ac} \over 2a}}\)
\(x = {{-b \over 2a }} ± {{\sqrt {(b^2 - 4ac) \over 2a }}} \)
\(x = {{{[-b ± \sqrt {(b^2 - 4ac)]} \over 2a}}} \)


In this case, the quadratic equation's “+” and “-” signs indicate different roots. In most cases, this intricate process is omitted, and the necessary roots are obtained solely by applying the quadratic formula.

How to Graph a Quadratic Equation

By expressing the quadratic equation as a function \(y = ax^2 + bx + c\), the graph of the quadratic equation \(ax^2 + bx + c = 0\) can be found. Additionally, we can obtain values of y by substituting different values of x, which gives us the required points to plot the graph. To create a parabola, these points can be displayed on the coordinate plane.


The quadratic equation's solutions are the x-values where its graph crosses the x-axis. By setting y = 0 in the function \(y = ax^2 + bx + c\) and solving for x, these points can be obtained algebraically.

Common Roots of Quadratic Equations

Let \(a_1x^2 + b_1x + c_1 = 0\) and \(a_2x^2 + b_2 x + c_2 = 0\) be two quadratic equations with common roots. To determine the conditions under which these two equations share a root, let's solve them. The two equations are solved for x2 and x.

\({(x^2)} {(b_1c_2 - b_2 c_2)} = {{{(-x)} \over {(a_1c_2 - a_2c_1)}}} = {{1} \over {(a_1b_2 - a_2b_1)}}\)
\(x^2 ={{ (b_1c_2 - b_2c_1) \over (a_1b_2 - a_2b_1) }} \)
\(x = {{(a_2c_1 - a_1c_2) \over (a_1b_2 - a_2b_1)}}\)

Therefore, the following condition for the two equations having a common root is obtained by simplifying the two expressions above.
\({{(a_1b_2 – a_2b_1) (b_1c_2 - b_2c_1) = (a_2c_1 - a_1c_2)^2}}\)

Maximum and Minimum Value of Quadratic Expression

Graphs can be used to represent the maximum and minimum values of a quadratic function in the form \(f(x) = ax^2 + bx + c\). If the value of a is positive (a > 0), then the parabola opens upwards and the maximum value of x is at \({{x = {{-b\over 2a}}}}\). It has a maximum value at \({x = {-b\over 2a}}\) for negative values of a (a < 0). And the parabola opens downwards.

The maximum and minimum values are used to determine the range of the functions:

The range is \([f{({-b\over 2a})}, ∞]\) for positive values of a (a > 0) 
The range is \([(-∞, f({-b\over 2a}))]\) for negative values of a (a < 0).

Range for a > 0: \([f{({-b\over 2a})}, ∞]\)
Range: \([(-∞, f({-b\over 2a}))]\) for a < 0.

Keep in mind that the domain of any quadratic function is all real numbers, or (-∞, ∞).
 

Tips and Tricks to Master Quadratic Equations

Mastering quadratic equations becomes easier with the right approach and consistent practice. These tips and tricks help students learn effectively while providing parents with simple, practical ways to guide and support their child’s learning.

  • Always start by identifying the value of a, b, and c in the equation \(ax^2 + bx + c = 0\). Knowing these values helps students to choose the best solving method such as factoring, completing the square, or using the quadratic formula.
     
  • Understand the pattern, as factoring becomes easy when you recognize these patterns. For example, \(x^2 + 5x + 6 = (x + 2)(x + 3)\).
     
  • Memorize the quadratic formula, that is \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) . It helps students to solve the quadratic equation easily. 
     
  • Use graph to visualize the quadratic equation, it helps to understand how it behaves. The point where the curve meets the x-axis are the solutions. 
     
  • To understand the number of roots, always check \(b^2 - 4ac\). If the value of discriminant is positive it has two real roots, zero it has one real root, and negative it has two complex roots.
     
  • Teachers can provide students with a quadratic equation worksheet with a mix of quadratic equation examples, including factoring, completing the square, and solving using the quadratic formula. 
     
  • Parents can show how quadratic equations appear in real life, such as projectile motion or area problems.
     
  • Teachers can break the problems into smaller steps by encouraging students to first rewrite the equation in standard form, identify coefficients, choose a method, and then solve.     

Common Mistakes and How to Avoid Them in Quadratic Equations

This section highlights common mistakes that students frequently make when working with quadratic equations, such as incorrect signs, poor factoring, or improper formula usage. It offers helpful guidance on how to steer clear of these pitfalls and approach problems with greater assurance. Some of the common mistakes are as follows:

Real-Life Applications of Quadratic Equations

Quadratic equations are not just abstract concepts in mathematics, but they play a crucial role in solving real-world problems across various fields. They are used to modeling situations where the relationship between variables forms a parabolic curve. Here are some practical applications:

  • Quadratic equations help determine the path of objects thrown or launched into the air. For instance, they are used to calculate the maximum height a basketball reaches or the distance a cricket ball travels.
  • In gaming and animation, quadratics are used to simulate realistic movements of characters and objects. This includes designing jumps, falls, or curved paths of moving elements.
  • Quadratic equations are applied to predict profits, losses, or investment growth when factors like interest rates, time, and principal amounts vary.
  • Engineers use quadratic equations to design parabolic arches, bridges, and structures, ensuring both strength and aesthetic appeal.
  • Quadratics are employed to calculate quantities like acceleration, displacement, and energy in systems where motion follows a curved trajectory.

Download Worksheets

Problem 1

Determine the solution of the Quadratic Equation: x² - 7x + 12 = 0

Okay, lets begin

x = 3 and x = 4.
 

Explanation

 The quadratic expression should be factored. We will understand it through a step-by-step explanation.

Step 1: Find two numbers that add up to -7 and multiply to 12.
Those are -3 and -4.
So, \(x^2 - 7x + 12\) 
= \(x^2 - 3x - 4x + 12\) 
= \((x - 3) (x - 4)\) (x will be constant)

Step 2: Use the zero-product property.
Set each factor to 0 after determining that \((x−3)(x−4)=0\):

        \( x - 3 = 0; \implies x = 3\). (We solve for 𝑥 by adding 3 to both sides of the equation.)

\(x - 4 = 0; \implies   x = 4 \)(we will add 4 to both sides).

Therefore, the final answer will be x = 3 and x = 4. 
 

Well explained 👍

Problem 2

Find the solution to the quadratic equation: x² + 4x + 8 = 0

Okay, lets begin

\({{x = -2 + 2i}} {\text { and }} {{x = -2 - 2i}}\)
 

Explanation

We will solve the equation following the steps:

Step 1: Determine the coefficients that will be, 
a = 1, b = 4, and c = 8

Step 2: Apply the quadratic formula:

       \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
\(x = {-4 \pm \sqrt{16 - 32} \over 2}\)

\(= {- 4 ± \sqrt {-16} \over 2}\)                           

Step 3: Make the square root simpler:

\(\sqrt {-16} = 4i\) = 4i  (the imaginary number \(i = \sqrt {(-1))}\)

\(x = {-4 ± 4i \over 2}\)

\(= -2 ± 2i\)

Therefore, the final answer is \(x = - 2 + 2i\) and \(x = - 2- 2i\).
 

Well explained 👍

Problem 3

The product of two consecutive positive numbers is 156. Find the numbers.

Okay, lets begin

12 and 13.
 

Explanation

We will be solving the quadratic word problems as well by following the steps.


Step 1: Let the numbers be x and x + 1.


Step 2: The equation will be formed as \(x(x + 1) = 156\)


Step 3: Expansion of the equation by moving all the terms to one side: 
                                   \(x^2 + x - 156 = 0\)             

Step 4: Calculate the quadratic factor and find two numbers that add up to one and multiply by -156.
13 and -12 are the numbers we’re looking for. So, \(13 \times -12 = -156\) and \(13 + (-12) = 1\).

Therefore, after solving the equation, we get \((x + 13)(x - 12) = 0\)

Step 5: Solve for the value of x:

\(x+13 = 0 \implies  x = – 13\) (which is not positive)
\( (x-12=0) \implies x=12 \)
After avoiding the negative sign, the numbers are 12 and 13.
 

Well explained 👍

Problem 4

Solve x^2-5x-6=0

Okay, lets begin

x = 6, and x = -1
 

Explanation

Step 1: Determine the standard form.

    The standard form of quadratic polynomial is: \(x^2 +bx + c= 0 \)
Given, \({x^2} - 5x - 6 = 0\)
Here, \(a =1, b = -5, c = -6\)

Step 2:  The quadratic should be factored. We search for two figures that:
To get the product, multiply \({a \cdot c} = {1 \cdot (-6)} = -6\)
To get the middle term, add \(b = -5\)

The numbers are -6 and 1 (because \(-6 \times 1 = -6\) and \(-6 + 1= -5\))
Therefore, 
                               \(x^2 - 5x - 6 = x^2 - 6x + x - 6\)
                              \(\implies x (x - 6) + 1(x - 6)\)
                               \(\implies (x - 6)(x + 1) = 0\)

Step 3: Finally, set each factor to 0.

  • \(x - 6 = 0 \implies x = 6 \)
  • \(x + 1 = 0 \implies x = -1 \)

Therefore, the final answer will be x = 6, x = -1.
 

Well explained 👍

Problem 5

Solve x²+ 6x + 9 = 0

Okay, lets begin

-3
 

Explanation

 After identifying the ideal square, this trinomial is considered a perfect square.

                         \(x^2 + 6x + 9 = (x + 3)^2\)

The second and final step involves factorization of the equation.

                         \((x + 3)^2 = 0\)
                          \(x + 3 = 0\)
                          \(x = -3\)
Therefore, the final answer will be x = -3 (can be a repeated or double root).  
 

Well explained 👍

FAQs on Quadratic Equations

1.What is a quadratic equation?

 A mathematical expression with a variable raised to the second power is called a quadratic equation. It is expressed as \(ax^2 + bx + c = 0\), where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 cannot be equal to zero. Since the highest power of the variable (typically 𝑥) in these equations is 2.

2.What is the discriminant, and what does it tell us?

The Discriminant is that part of the quadratic formula that helps in identifying the type of solutions required for a quadratic equation. It can be found by substituting the values in the equation \(b^2 − 4ac\). It is important to find the value of the discriminant because it tells us whether the solutions are real or complex, and whether they are repeated or distinct.
 

3.What is the vertex of a quadratic function?

Depending on which way the graph opens, the vertex of a quadratic function is the parabola's turning point, which can be either the maximum or minimum value. It is the location where the curve shifts course. The formula, \(x= {{-b\over 2a}}\), can be used to find the x-value of the vertex of any quadratic function expressed in standard form. The corresponding y-value can then be obtained by plugging this value back into the equation.

4.How does a quadratic equation's graph appear?

A parabola is a curved shape that represents the graph of a quadratic equation. The sign of the leading coefficient (𝑎) determines whether this parabola opens upward or downward. The graph opens upward in the form of a "U" if 𝑎 is positive and downward in the form of an upside-down "U" if 𝑎 is negative. The graph is always symmetrical, and the vertex denotes the highest or lowest point.
 

5.Is it possible to factor all quadratic equations?

Simple integers are not always sufficient to factor all quadratic equations. Certain quadratics necessitate the application of more sophisticated solving techniques, such as the quadratic formula or completing the square, particularly when the roots of the equation are complex or irrational. When the equation has neat, integer solutions, factoring performs best. The other approaches are trustworthy substitutes for factoring if it proves ineffective.
 

6.What are the methods to solve quadratic equations?

There are different methods to solve quadratic equations like: factoring, completing the square, quadratic formula, and graphing. 

7.What real-life examples can I give my child?

To help students to learn the quadratic equations, parents can connect it with real life examples, such as: 

  • The height of a thrown ball or a cricket ball's trajectory. 
  • Calculating areas of rectangular or square plots with changing dimensions. 
  • Predicting profit in a business scenario. 

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.