#### Question: Find the roots of the following equations:

i. x - \\frac{1}{x}) = 3, x ≠ 0

ii. \\frac{1}{x + 4}) - \\frac{1}{x - 7}) = \\frac{11}{30}), x ≠ -4, 7

#### Video Explanation

#### Explanatory Answer

##### i. x - \\frac{1}{x}) = 3, x ≠ 0

Because x ≠ 0, multiply both sides of the equation by x.

x^{2} - 1 = 3x

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

We cannot factorize this equation to find roots. So, let us use the quadratic formula and find the roots of this equation.

The roots of ax^{2} + bx + c = 0 are \\frac {-b\pm \sqrt {{b}^{2}-4ac}} {2a})

In this equation, a = 1, b = -3 and c = -1

b^{2} – 4ac = (-3)^{2} - 4 * 1 * (-1) = 9 + 4 = 13

∴ x = \\frac{-(-3)士 {\sqrt{13}}}{2}) = \\frac{3 士 {\sqrt{13}}}{2})

x = \\frac{3 + {\sqrt{13}}}{2}) and \\frac{3 - {\sqrt{13}}}{2})

##### ii. \\frac{1}{x + 4}) - \\frac{1}{x - 7}) = \\frac{11}{30}), x ≠ -4, 7

Because x ≠ -4, 7 multiply the equation by (x + 4) (x - 7)

(x - 7) - (x + 4) = \\frac{11}{30})(x + 4) (x - 7)

-11 = \\frac{11}{30})(x + 4) (x - 7)

Multiply the equation by \\frac{30}{11})

-30 = (x + 4) (x - 7)

= x^{2} - 7x + 4x - 28 + 30 = 0

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

##### Method 1: Factorize the quadratic equation to find the values of x

x^{2} - 2x - x + 2 = x(x - 2) - 1 (x - 2) = 0

(x - 2 )(x - 1) = 0

x = 2 and x = 1

##### Method 2: Use the quadratic formula to find the values of x

The roots of ax^{2}+ bx + c = 0 are \\frac {-b\pm \sqrt {{b}^{2}-4ac}} {2a})

In this equation, a = 1, b = -3 and c = 2

b^{2} – 4ac = (-3)^{2} – 4 * 1 * 2 = 9 – 8 = 1 > 0

Therefore, x = -\\frac{-(-3)士{\sqrt1}}{2}) = \\frac{3 士 1}{2})

x= 2 and x = 1