Check whether the following are quadratic equations: (i) (x - 2)2 + 1 = 2x - 3 (ii) x (x + 1) + 8 = (x + 2) (x - 2) (iii) x (2x + 3) = x2 + 1 (iv) (x + 2)3 = x3 - 4

Solution:

A quadratic equation is in the form of ax2 + bx + c = 0.

(i) (x - 2)2 + 1 = 2x - 3

Using the algebraic identity (a - b)2 = a2 - 2ab + b2

⇒ (x - 2)2 + 1 = 2x - 3 can be rewritten as

⇒ x2 - 4x + 5 = 2x - 3

⇒ x2 - 6x + 8 = 0

∵ It is of the form ax2 + bx + c = 0.

∴ The equation is a quadratic equation.

ii) x(x + 1) + 8 = x2 + x + 8

and (x + 2)(x - 2) = x2 - 4 [ ∵ (a + b) (a - b) = a2 - b2 ]

⇒ x2 + x + 8 = x2 - 4

⇒ x + 12 = 0 

∵ It is not of the form ax2 + bx + c = 0

∴ The equation is not a quadratic equation

iii) x (2x + 3) = x2 + 1 

⇒ 2x2 + 3x = x2 + 1

Subtracting x2 from both sides we get.

x2 + 3x = 1

Subtracting 1 from both sides, we get

x2 + 3x - 1 = 0 

∵ It is of the form ax2 + bx + c = 0.

∴ The equation is a quadratic equation.

iv) (x + 2)3 = x3 - 4 

Using the algebraic identity (a + b)3 = a3 + 3a2b+ 3ab2 + b3

⇒ (x + 2)3 = x3 - 4 

⇒ x3 + 6x2 + 12x + 8 = x3 - 4

Subtracting x3 from both sides, we get

⇒ 6x2 + 12x + 8 = - 4

Adding 4 to both sides, we get

⇒ 6x2 + 12x + 12 = 0

2 (x2 + 2x + 2) = 0 

x2 + 2x + 2 = 0

∵ It is of the form ax2 + bx + c = 0. 

∴ The equation is a quadratic equation

☛ Check: NCERT Solutions for Class 10 Maths Chapter 4

Check whether the following are quadratic equations: (i) (x - 2)2 + 1 = 2x - 3 (ii) x (x + 1) + 8 = (x + 2) (x - 2) (iii) x (2x + 3) = x2 + 1 (iv) (x + 2)3 = x3 - 4

Summary:

On simplifying the equations, (i) (x - 2)2 + 1 = 2x - 3, (iii) x (2x + 3) = x2 + 1 and (iv) (x + 2)3 = x3 - 4 are of the form ax2 + bx + c = 0, thus are quadratic equation but (ii) x(x + 1) + 8 = (x + 2) (x - 2) is not of the form ax2 + bx + c = 0, so it is not a quadratic equation

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