Check whether the following are quadratic equations

Q1: Check whether the following are quadratic equations:

(i)   (x + 1)² = 2(x – 3)

(ii)  x² – 2x = (–2) (3 – x)

(iii) (x – 2)(x + 1) = (x – 1)(x + 3)

(iv) (x – 3)(2x +1) = x(x + 5)

(v)  (2x – 1)(x – 3) = (x + 5)(x – 1)

(vi) x² + 3x + 1 = (x – 2)²

Answer:

 (i)   (x + 1)² = 2(x – 3)

use formula (a+b)²=a² +2ab +b²

or,  x² + 2x + 1 = 3x – 6

subtract 3x and add 6 both side we get

or,  x² + 2x -3x + 1 + 6 = 0

or,  x² - x + 7 = 0

Above equation is in form of ax² + bx + c = 0, so it is a quadratic equation.

(ii) x² – 2x = (–2) (3 – x)

or, x²– 2x = –6 + 2x

or,  x²– 2x - 2x + 6 = 0

or,  x²– 4x + 6 = 0

Above equation is in form of ax² + bx + c = 0, it is a quadratic equation.

(iii) (x – 2)(x + 1) = (x – 1)(x + 3)

Open the bracket

or, x² + x -2x - 2 = x² + 3x -x -3

simplify it we get

or, x² - x - 2 = x² + 2x - 3

or, x² - x - 2 - x² - 2x + 3 = 0

or, -3x + 1 = 0

change the sign

 or, 3x - 1 = 0

Above equation is NOT in form of ax² + bx + c = 0, it is NOT a quadratic equation.

(iv) (x – 3)(2x +1) = x(x + 5)

Open the bracket

or, 2x² + x - 6x - 3 = x² + 5x

Simplify it

or, 2x² - 5x - 3 -  x² - 5x = 0

subtract the values

or, x²- 10x - 3 = 0

Above equation is in form of ax² + bx + c = 0, it is a quadratic equation.

(v)  (2x – 1)(x – 3) = (x + 5)(x – 1)

Open the bracket

or, 2x² -  6x  - 1x + 3 = x² - 1x + 5x - 5

simplify it

or, 2x² - 7x + 3 = x²+ 4x - 5

subtract the values

or, 2x² - 7x + 3 - x² - 4x + 5 = 0

simplify it again

or, x² - 11x + 8 = 0

Above equation is in form of ax² + bx + c = 0, it is a quadratic equation.

(vi) x² + 3x + 1 = (x – 2)²

use formula (a-b)²=a² -2ab +b²

or, x² + 3x + 1 = x² + 4 - 4x

subtract the value

or, x² + 3x + 1 - x² - 4 + 4x =0

simplify it

or, 7x - 3 = 0

Above equation is NOT in form of ax² + bx + c = 0, it is NOT a quadratic equation.