QUADRATIC EQUATIONS An equation obtained by equating a

QUADRATIC EQUATIONS An equation obtained by equating a quadratic polynomial to zero is called a quadratic equation. Any equation of the form ax 2+ bx +c = 0 where a, b, c are real numbers and a ≠ 0 is called a quadratic equation in variable x.

Standard form of a quadratic equation The standard form of a quadratic equation in variable x is ax 2+ bx +c = 0 where a , b, c are real numbers and a ≠ 0. Example: 3 x 2+ 2 x +1 = 0 x 2 - 4 = 0

To check whether the given equations are quadratic or not. Exercise 4. 1 Q 1. Check whether the following are quadratic equations: ( i ) (x+1)2 = 2 (x-3) x 2 + 2 x + 1 = 2 x – 6 x 2 + 2 x + 1 - 2 x + 6 = 0 x 2 + 7 = 0 x 2 +0. x + 7 = 0 It is of the form ax 2+ bx +c = 0 with a ≠ 0. Hence the given equation is a quadratic equation.

( ii) x 2 – 2 x = (-2) (3 – x) x 2 – 2 x = -6 + 2 x x 2 – 2 x + 6 - 2 x = 0 x 2 – 4 x + 6 = 0 It is of the form ax 2+ bx +c = 0 with a ≠ 0. Hence the given equation is a quadratic equation.

(iii) (x – 2) (x + 1) = (x - 1) (x +3) x 2 + x -2 = x 2 + 3 x – x - 3 x 2 – x - 2 = x 2 + 2 x -3 x 2 – x - 2 - x 2 - 2 x + 3 = 0 -3 x +1 = 0 It is not of the form ax 2+ bx +c = 0 with a ≠ 0. Hence the given equation is not a quadratic equation.

(iv) (x-3)(2 x+1) = x (x+5) 2 x 2 + x – 6 x - 3 = x 2 + 5 x 2 x 2 + x – 6 x - 3 - x 2 - 5 x = 0 x 2 - 10 x – 3 = 0 It is of the form ax 2+ bx +c = 0 with a ≠ 0. Hence the given equation is a quadratic equation.

(v) (2 x-1) (x-3) = (x+5) (x-1) 2 x 2 - 6 x - x + 3 = x 2 – x + 5 x -5 2 x 2 - 7 x + 3 = x 2 + 4 x -5 2 x 2 - 7 x + 3 - x 2 - 4 x + 5 = 0 x 2 - 11 x + 8 = 0 It is of the form ax 2+ bx +c = 0 with a ≠ 0. Hence the given equation is a quadratic equation.

(vi) x 2 +3 x + 1 = (x- 2)2 x 2 +3 x + 1 = x 2 - 4 x + 4 x 2 +3 x + 1 - x 2 + 4 x - 4 = 0 7 x – 3 = 0 It is not of the form ax 2+ bx +c = 0 with a ≠ 0. Hence the given equation is not a quadratic equation.

(vii) (x +2)3 = 2 x (x 2 – 1) x 3 + 6 x 2 +12 x + 8 = 2 x 3 – 2 x x 3 + 6 x 2 +12 x + 8 - 2 x 3 + 2 x = 0 - x 3 + 6 x 2 + 14 x + 8 = 0 It is not of the form ax 2+ bx +c = 0 with a ≠ 0. Hence the given equation is not a quadratic equation.

(viii) x 3 - 4 x 2 - x + 1 = (x -2)3 x 3 - 4 x 2 - x + 1 = x 3 - 6 x 2 + 12 x - 8 x 3 - 4 x 2 - x + 1 - x 3 + 6 x 2 - 12 x + 8 = 0 2 x 2 - 13 x + 9 = 0 It is of the form ax 2+ bx +c = 0 with a ≠ 0. Hence the given equation is a quadratic equation