### Question 1. Write the following quadratic equations in the standard form and point out pure quadratic equations.

**(i). (x + 7)(x − 3) = −7**

### Solution:

The given quadratic equation is

* (x + 7)(x − 3) = −7* (i)

Multiplying the expressions (x + 7) and (x *−* 3) in equation (i) with each other, we have

* x ^{2} + 7x − 3x − 21 = − 7
*

*⇒ x ^{2} + 4x − 21 = − 7
*

Now adding 7 on both sides of the above equation and get

* x ^{2} + 4x − 21 + 7 = − 7 + 7
*

*⇒ x^{2} + 4x − 14 = 0 (ii)
*

Equation (ii) is in the standard form of the quadratic equation but it is not a pure quadratic equation because b ≠ 0.

**(ii). (x^{2} + 4)/3 − x/7 = 1**

### Solution:

The given quadratic equation is

* (x ^{2} + 4)/3 − x/7 = 1* (i)

By solving the expression *(x ^{2} + 4)/3 − x/7* in equation (i), we have

* (7x ^{2} + 28 − 3x)/21 = 1
*

Now multiplying both sides of the above equation by 21, we get

* 7x ^{2} + 28 − 3x = 21*

*⇒ 7x ^{2} − 3x + 28 = 21
*

Now subtracting 21 from both sides of the above equation and get

* 7x ^{2} − 3x + 28 − 21 = 21 − 21
*

*⇒ 7x^{2} − 3x + 7 = 0 (ii)
*

Equation (ii) is in the standard form of the quadratic equation but it is not a pure quadratic equation because b ≠ 0.

**(iii). x/(x + 1) + (x + 1)/x = 6**

### Solution:

The given quadratic equation is

* x/(x + 1) + (x + 1)/x = 6* (i)

By solving the expression * x/(x + 1) + (x + 1)/x* in equation (i), we have

* x ^{2} + x^{2} + 2x + 1 / x(x + 1) = 6*

*⇒ 2x ^{2} + 2x + 1 / x^{2} + x = 6*

Now multiplying both sides of the above equation by *x ^{2} + x*, we get

* 2x ^{2} + 2x + 1 = 6(x^{2} + x)*

*⇒ 2x ^{2} + 2x + 1 = 6x^{2} + 6x*

*⇒ 6x ^{2} − 2x^{2} + 6x − 2x − 1 = 0*

*⇒ 4x^{2} + 4x − 1 = 0 (ii)
*

Equation (ii) is in the standard form of the quadratic equation but it is not a pure quadratic equation because b ≠ 0.