Quadratic Equations | Roots, Formula, Solved Examples

Quadratic equations is an important topic in mathematics. A quadratic equation is an equation with degree two. I.e., it has at least one variable with a degree two (x² or y²). The term "quadratic" comes from the Latin word quadratus (meaning square), because the variable is squared. Let’s understand in detail about the quadratic equations along with definitions, formulas,  and examples. We will also learn how to find the roots of a quadratic equation with help of solved examples.

Table of Contents

• What is a Quadratic Equation
• Quadratic Equation Formula
• Proof for Quadratic Equation Formula
• Solution of a Quadratic Equation by Factorisation
• Solved Examples for Quadratic Equations
• Frequently Asked Questions on Quadratic Equations

What is a Quadratic Equation

A quadratic equation is a polynomial equation of degree two. The general form of a quadratic equation is f(x) = ax² + bx + c. In other words, we can explain the quadratic equation as a second-degree polynomial equation in a single variable, written in the standard form as: ax² + bx + c = 0,

where a ≠ 0 and a, b, c are real numbers and x is the variable.

Read more: Class 9 - Zeroes of Polynomial

Quadratic Equation Formula

Solving quadratic equations implies finding the roots of a quadratic equation. The roots or zeros of a quadratic equation are the values of x that make the equation equal to zero. In standard form a quadratic equation is written as: ax² + bx + c = 0, where a ≠ 0 and the general formula to find the roots of a quadratic equation is:

x = (−b ± √(b² − 4ac)) / 2a

The same formula can also be represented as: x = (-b ± √D) / (2a), where, D = b² - 4ac

In this formula, b² - 4ac is called discriminant (D)

Keynotes:

• The standard form of a quadratic equation is: ax² + bx + c = 0
• In a quadratic equation, ax² is the quadratic term, bx is the linear term and c is the constant term.
• A quadratic equation can have a maximum of two roots.
  
  

Proof for Quadratic Equation Formula

For a quadratic equation ax² + bx + c = 0, the roots are: $x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Let's prove it by completing the square
Given: ax² + bx + c = 0

Step 1: Divide throughout by a: $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$

Step 2: Move the constant to the right side:  $x^2+\frac{bx}{a}=\frac{−c}{a}​$

Step 3: Add  $(\frac{b}{2a})^2$ to both sides:  $x^2+\frac{ax}{b} +(\frac{b}{ax})^2=\frac{−c}{a}​ + (\frac{b}{ax})^2$ 

Step 4: Write the left side as a perfect square:  $(x + \frac{b}{2a})^2$=  $\frac{- c}{a} + \frac{b^2}{4a^2}$

Step 5: Simplify the right side: $(x + \frac{b}{2a})^2 = \frac{-4ac + b^2}{4a^2} = \frac{b^2 - 4ac}{4a^2}$

Step 6: Take square root on both sides: $x + \frac{b}{2a} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}}$ =  $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}$

Step 7: Find the value of x:  $x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$ =

The two roots are:  $\frac{-b + \sqrt{b^2 - 4ac}}{2a}$ and  $\frac{-b - \sqrt{b^2 - 4ac}}{2a}$

 

Solution of a Quadratic Equation by Factorisation

Solving any quadratic equation using factorisation involves splitting the middle term. Let's understand the step-by-step solution of a quadratic equation using factorization process:

• First split the middle-term by multiplying the coefficients of first and last as: a x c
• Then, finding two numbers m and n such that:
• m x n = a x c
• m + n = b
• The next step is to write the middle term bx as: mx + nx → ax² + mx +nx + c = 0
• Then, factorise by grouping terms as linear factors.
• Set each term equal to zero and solve for the value of x.

In general if a quadratic equation factorises into the form: (px + q)(rx + s) = 0, then:

Roots are x = -q/p or x = -s/r.

Solved Examples for Quadratic Equations

Example 1: Find the roots of x² + 7x + 10 = 0 using factorisation.

Solution 1: The given equation is x² + 7x + 10 = 0

Here, a = 1, b = 7 and c = 10.

By multiplying the first and last term coefficient we get, a × c = 1 × 10 = 10.

Now, find numbers that multiply to 10 and add to 7 → (5, 2).

Then, split the middle term as: x² + 5x + 2x + 10 = 0.

Group the terms as: (x² + 5x) + (2x + 10) = 0.

Write factors: x(x + 5) + 2(x + 5) = 0.

Take out the common factors to equate with zero and find the value of x: (x + 5)(x + 2) = 0.

Therefore, x = – 5 or x = –2

Example 2: Solve 2x² - 7x + 3 = 0 using quadratic formula.

Solution: Step 1: Identify coefficients in the given equation

Here, a = 2, b = -7, c = 3

Step 2: Find the discriminant (D)

D = b² - 4ac

= (-7)² - 4(2)(3)

= 49 - 24

= 25

Step 3: Apply the quadratic formula

x = (-b ± √D) / (2a)

x = (7 ± √25) / 4

x = (7 ± 5) / 4

Step 4: Calculate the roots

x = (7 + 5)/4 = 12/4 = 3

x = (7 - 5)/4 = 2/4 = 1/2