Chapter 4

Introduction to Quadratic Equations

Quadratic equation: An equation of the form ax²+ bx + c = 0 is called a quadratic equation, where a, b and c are known values (i.e. constants), a is non-zero and x is the unknown value (i.e. a variable).
e.g. 5x²+ 7x + 3 = 0, 4x²+ 2 = 0, 3x²+ 8x + 4 = 8, are all quadratic equations.

Since the highest power of a quadratic equation is 2, a quadratic equation can have at the most two unique solutions, which are also called the roots of the equation. Thus, for a given quadratic equation, x can have at most 2 unique values.

Solution of a quadratic equation: x =  is a solution of the quadratic equation, or that  satisfies the quadratic equation.

The zeroes of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are the same.

Solution of a quadratic equation by factorisation: We find the roots of ax² + bx + c = 0 by factorizing ax² + bx + c into two linear factors and equating each factor to zero.

Solution of a Quadratic Equation by Completing the Square: Consider the quadratic equation ax² + bx + c = 0 (a ≠ 0).

Dividing throughout by a, we get

So, the roots of the given equation are the same as those of

If b² – 4ac ≥ 0, then by taking the square roots in (i), we get

Thus, if b² – 4ac ≥ 0, then the roots of the quadratic equation ax² + bx + c = 0 are given by.

This formula for finding the roots of a quadratic equation is known as the “Quadratic formula.”

Discriminant: b² – 4ac determines whether the quadratic equation ax² + bx + c = 0 has real roots or not, b² – 4ac is called the discriminant of this quadratic equation.

Nature of Roots:

(i) If b² – 4ac > 0, we get two distinct real roots.

(ii) If b² – 4ac = 0, then x =

 x = or.

(iii) If b² – 4ac < 0, then there is no real number whose square is b² – 4ac. Therefore, there are no real roots for the given quadratic equation in this case.