Quadratic Equations

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NCERT Revision Notes for Chapter 4 Quadratic Equations Class 10 Maths

CBSE NCERT Revision Notes

1

Introduction to Quadratic Equations

Answer

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

A quadratric polynomials when equated to zero can form quadratic Equation. 

Any Equation which can be exprssed 

2

Roots of a Quadratic Equation

Answer

Roots of a quadratic Equation is similar to the zeros of quadratic polynomial. 

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 ot both roots may be Equal.

3

Methods for Solving Quadratic Equation - By Factorisation

Answer

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

4

Methods for Solving Quadratic Equation - By Completing Square

Answer

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

5

Methods for Solving Quadratic Equation - By Quadratic Formula

Answer

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

6

Discriminant

Answer

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 the quadratic Formula as 

 

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

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