Roots of Quadratic Equation

A quadratic equation is an algebraic equation whose degree is two. We shall learn how to find the roots of quadratic equations algebraically and using the quadratic formula. The general form of a quadratic equation is ax² + bx + c = 0, where x is the unknown and a, b and c are known quantities such that a ≠ 0. Since the degree of such an equation is two, we get two roots of any given quadratic equation.

Learn more about quadratic equations.

In this article, we shall learn and practice solving a quadratic equation by factoring the given equation and using the quadratic formula. Also, learn how to find the nature of the roots of a quadratic equation.

What is meant by the Roots of a Quadratic Equation?

Roots of a quadratic equation mean those values of the unknown for which the given equation equals zero. Let ax² + bx + c = 0 is a quadratic equation (a ≠ 0), if for x = 𝛼 and x = 𝛽 the given function vanishes. We say 𝛼 and 𝛽 are the roots of the quadratic equation and (x – 𝛼) and (x – 𝛽) are the factors of the quadratic equation.

Graphically, the roots of a quadratic equation mean those points where the curve of the given quadratic equation intersects the x-axis.

How to Find Roots of Quadratic Equation

Now, we shall discuss how to find the roots of a quadratic equation by different methods.

Completing the Squares Method

We shall reduce the given quadratic equation into any of the standard algebraic identities and then find the root of the equation. Let us take an example 2x² – 5x – 3 = 0

to understand this method.

2x² – 5x – 3 = 0

⇒ x² – (5/2)x – 3/2 = 0 (Dividing both sides by coefficient of x²)

⇒ x² – 2. ½ . 5/2. x – 3/2 = 0

[adjusting the given quadratic equation to reduce it into the form (a – b)²]

⇒ x² – (2.5)/(4.x) – 3/2 = 0

⇒ x² – (2.5)/(4.x) + 25/16 – 3/2 = 25/16 (adding both sides by 25/16)

⇒ x² – 2.(5/4).x + (5/4)² = 25/16 + 3/2

⇒ (x – 5/4)² = 49/16

⇒ x – 5/4 = ± 7/4

⇒ x = 5/4 ± 7/4

∴ x = 5/4 + 7/4 and x = 5/4 – 7/4

⇒ x = 3 and x = –½

Split the Middle Term Method

In this method we find the roots of the given quadratic equation ax² + bx + c = 0 by splitting the coefficient of x. We have to find two numbers 𝛼 and 𝛽 such that 𝛼 + 𝛽 = b and 𝛼𝛽 = c. For example, we have to find the roots of x² – 5x + 6 = 0

• 6 = 2 × 3 and –5 = –3 –2
• We shall split the middle term as: x² –3x –2x + 6 = 0
• Factorising: x(x –3) –2 (x – 3) = 0 ⇒ (x –2)(x – 3) = 0
• ∴ x = 2 and 3

Using Quadratic Equation

For any given quadratic equation ax² + bx + c = 0, the roots of the equation are given by the following formula:

For example, for the quadratic equation x² – 5x + 3 = 0

∴ x = [5 + √13]/2 and [5 – √13]/2

Nature of Roots of Quadratic Equation

The nature of the roots of a quadratic equation is determined by finding the value of the discriminant D.

For the quadratic equation ax² + bx + c = 0; a ≠ 0

Discriminant D = b² – 4ac

Learn more about nature of roots of quadratic equation.

Video Lessons

Visualising square roots

Finding Square roots

Related Articles

• Quadratic Equations for Class 10
• Quadratic Formula
• Factorisation
• Polynomials

Solved Examples on Roots of Quadratic Equations

Example 1:

Find the roots of the following quadratic equation using completing the square method:

x² – 3x – 10 = 0.

Solution:

We have x² – 3x – 10 = 0

Multiplying and dividing the middle term by 2, we get

⇒ x² – 2. ½. 3x – 10 = 0

Adding both sides by 9/4, we get

⇒ x² – 2.(3/2).x – 10 + 9/4 = 9/4

⇒ x² – 2.(3/2).x + 9/4 = 10 + 9/4

⇒ (x – 3/2)² = 49/4 {∴ (a + b)² = a² + 2ab + b²}

Taking square roots on both the sides, we get

⇒ x – 3/2 = ±7/2

⇒ x = 7/2 + 3/2 and –7/2 + 3/2

⇒ x = 5 and –2

Example 2:

Find the roots of the quadratic equation: x² – 8x + 15 = 0

Solution:

We shall use the quadratic formula,

⇒ x = (8 + 2)/2 and (8 – 2)/2

⇒ x = 5 and 3.