i.e., x 2 – 6x + 8 = 0

It is in the form of ax 2 + bx + c = 0.

Therefore, the given equation is a quadratic equation.

ii.      Here LHS = x(x + 1) + 8 = x 2 + x + 8

and RHS = (x + 2)(x – 2) = x 2 – 4

Therefore, x 2 + x + 8 = x 2 – 4

x 2 + x + 8 - x 2 + 4 = 0

i.e., x + 12 = 0

It is not in the form of ax 2 + bx + c = 0, (a ¹ 0)

Therefore, the given equation is not a quadratic equation.

iii.     Here, LHS = x (2x + 3) = 2x 2 + 3x

So, x (2x + 3) = x 2 + 1 can be rewritten as

2x 2 + 3x = x 2 + 1

Therefore, we get x 2 + 3x – 1 = 0

It is in the form of ax 2 + bx + c = 0.

So, the given equation is a quadratic equation.

iv.     Here, LHS = (x + 2)3 = (x + 2)2 (x + 2)

= (x 2 + 4x + 4) (x + 2)

= x 3 + 2x 2 + 4x 2 + 8x + 4x + 8

= x 3 + 6x 2 + 12x + 8

Therefore, (x + 2)3 = x 3 – 4 can be rewritten as

x 3 + 6x 2 + 12x + 8 = x 3 – 4

i.e.,6x 2 + 12x + 12 = 0

or, x 2 + 2x + 2 = 0

It is in the form of ax 2 + bx + c = 0.

So, the given equation is a quadratic equation.

Remark : In (ii) above, the given equation appears to be a quadratic equation, but it is not a quadratic equation.

    In (iv) above, the given equation appears to be a cubic equation (an equation of degree 3) and not a quadratic equation.

But it turns out to be a quadratic equation.

As you can see, often we need to simplify the given equation before deciding whether it is quadratic or not.

page no:109

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