Notes of Ch 4 Quadratic Equations | Class 10th Maths

Study Materials and Revision Notes for Ch 4 Quadratic Equations Class 10th Maths

• Quadratic Equations
• Roots of a Quadratic Equation
• Number Roots
• Methods for solving Quadratic Equation
• Discriminant
• Nature of Roots

Quadratic Equations: A quadratic equation is in the form of ax²+ bx + c = 0 where a, b, c are real numbers and a ≠ 0.
Example: (i) 3x² + 4y - 6 = 0
(ii) 2x² + 6 = 0

Roots of a Quadratic Equation

Let ax² + bx + c = 0, be a quadratic equation. If α is a root of this equation. It means x = α satisfies this equation i.e., aα² + bα + c = 0

Number of Roots

A quadratic equation has two roots. The roots may or may not be real.

Methods For Solving Quadratic Equation 

There are three methods for solving quadratic equation:
(a) By factorization
(b) By completing the square
(c) By Quadratic Formula

By Factorization

Only real roots are found by factorization. If we can factorize ax² + bx + c = 0, where a ≠ 0, into a product of two linear factors, then the roots of this quadratic equation can be calculated by equating each factor to zero.

For example:
x² – 3x – 10 = 0
⇒ x² - 5x + 2x - 10 = 0
⇒ x(x - 5) + 2(x - 5) = 0
⇒ (x - 5)(x + 2) = 0
Roots of this equation are the values for which (x - 5)(x + 2) = 0

∴ x - 5 = 0 or x + 2 = 0

⇒ x = 5 or x = -2

By completing the square method

A quadratic equation can also be solved by the method of completing the square. We will try to convert the given equation in the form of (a+b)² or (a-b)².

For example:

2x² – 7x + 3 = 0
⇒ 2x² – 7x = - 3
On dividing both sides of the equation by 2, we get

⇒ x² – 7x/2  = -3/2

⇒ x² – 2 × x ×  7/4 = -3/2
On adding (7/4)² to both sides of equation, we get
⇒ (x)² - 2 × x × 7/4 + (7/4)² = (7/4)² - 3/2

⇒ (x - 7/4)² = 49/16 - 3/2
⇒ (x - 7/4)² = 25/16
⇒ (x - 7/4) = ± 5/4
⇒ x = 7/4 ± 5/4
⇒ x = 7/4 + 5/4 or x = 7/4 - 5/4
⇒ x = 12/4 or x = 2/4
⇒ x = 3 or 1/2

Quadratic Formula

To find roots of ax² + bx + c = 0 is given by

We will solve a question using quadratic formula.

4x² + 4√3x + 3 = 0
On comparing this equation with ax² + bx + c = 0, we get
a = 4, b = 4√3 and c = 3
By using quadratic formula, we get
x = -b±√b² - 4ac/2a
⇒ x = -4√3±√48-48/8
⇒ x = -4√3±0/8
∴ x = -√3/2 or x = -√3/2

Discriminant

For the quadratic equation ax² + bx + c = 0 the expression is called the discriminant and denoted by D. Then the roots of the quadratic equation are given by 

Nature of roots 

The nature of roots for ax² + bx + c = 0 can be determined through discriminant.
(i) If D 0, then roots are real and unequal.
(ii) D=0, then the equation has equal and real roots.
(iii) D<0, then the equation has no real roots