# 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) = x^{2} + 1 (iv) (x + 2)^{3} = x^{3} - 4

**Solution:**

A quadratic equation is in the form of ax^{2} + bx + c = 0.

(i) (x - 2)^{2} + 1 = 2x - 3

Using the algebraic identity (a - b)^{2} = a^{2} - 2ab + b^{2}

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

⇒ x^{2} - 4x + 5 = 2x - 3

⇒ x^{2} - 6x + 8 = 0

∵ It is of the form ax^{2} + bx + c = 0.

∴ The equation is a quadratic equation.

ii) x(x + 1) + 8 = x^{2} + x + 8

and (x + 2)(x - 2) = x^{2} - 4 [ ∵ (a + b) (a - b) = a^{2} - b^{2} ]

⇒ x^{2} + x + 8 = x^{2} - 4

⇒ x + 12 = 0

∵ It is not of the form ax^{2} + bx + c = 0

∴ The equation is not a quadratic equation

iii) x (2x + 3) = x^{2} + 1

⇒ 2x^{2} + 3x = x^{2} + 1

Subtracting x^{2} from both sides we get.

x^{2} + 3x = 1

Subtracting 1 from both sides, we get

x^{2} + 3x - 1 = 0

∵ It is of the form ax^{2} + bx + c = 0.

∴ The equation is a quadratic equation.

iv) (x + 2)^{3} = x^{3} - 4

Using the algebraic identity (a + b)^{3} = a^{3} + 3a^{2}b+ 3ab^{2} + b^{3}

⇒ (x + 2)^{3} = x^{3} - 4

⇒ x^{3} + 6x^{2} + 12x + 8 = x^{3} - 4

Subtracting x^{3} from both sides, we get

⇒ 6x^{2} + 12x + 8 = - 4

Adding 4 to both sides, we get

⇒ 6x^{2} + 12x + 12 = 0

2 (x^{2} + 2x + 2) = 0

x^{2} + 2x + 2 = 0

∵ It is of the form ax^{2} + 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) = x^{2} + 1 (iv) (x + 2)^{3} = x^{3} - 4

**Summary:**

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

**☛ Related Questions:**

- Represent the following situations mathematically: (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with...
- Find the roots of the equation 2x
^{2}- 5x + 3 = 0, by factorisation - Find the roots of the quadratic equation 6x
^{2}- x - 2 = 0

visual curriculum