3 Tutor System
Starting just at 265/hour

# Find the roots of the following equations: (i) $${{x} - {{1}\over {x}}} = 3, x \neq 0$$ (ii) $${{1} \over {x + 4}} - {{1} \over {x - 7}} = {{11} \over {30}}, x \neq -4,7$$

(i)$${{x} - {{1}\over {x}}} = 3$$ where x is not equal to 0

$$\Rightarrow$$ $${{x^2 - 1} \over {x}} = 3$$
$$\Rightarrow$$$$x^2 - 1 = 3x$$
$$\Rightarrow$$$$x^2 -3x - 1 = 0$$

The general form of equation is $$ax^2 + bx + c = 0$$

Quadratic formula = $$x = {{-b ± \sqrt{b^2 - 4ac}} \over {2a}}$$

$$\Rightarrow$$$$x = {{3 ± \sqrt{(3)^2 - 4(1)(-1)}} \over {(2)(1)}}$$
$$\Rightarrow$$$$x = {{3 ± \sqrt{13}} \over {2}}$$
$$\Rightarrow$$$$x = {{3 + \sqrt{13}} \over {2}} , {{3 - \sqrt{13}} \over {2}}$$

(ii)$${{1} \over {x + 4}} - {{1} \over {x - 7}} = {{11} \over {30}}, x ? -4,7$$
$$\Rightarrow$$$${{(x - 7) - (x + 4)} \over {(x - 4)(x - 7)}} = {{11} \over {30}}$$
$$\Rightarrow$$$${{-11} \over {(x - 4)(x - 7)}} = {{11} \over {30}}$$
$$\Rightarrow$$$$-30 = x^2 - 7x + 4x -28$$
$$\Rightarrow$$$$x^2 -3x + 2 = 0$$

The general form of equation is $$ax^2 + bx + c = 0$$

Quadratic formula = $$x = {{-b ± \sqrt{b^2 - 4ac}} \over {2a}}$$

$$\Rightarrow$$$$x = {{3 ± \sqrt{(3)^2 - 4(1)(2)}} \over {(2)(1)}}$$
$$\Rightarrow$$$$x = {{3 ± \sqrt{1}} \over {2}}$$
$$\Rightarrow$$$$x = {{3 + \sqrt{1}} \over {2}} , {{3 - \sqrt{1}} \over {2}}$$
$$\Rightarrow$$$$x = 2,1$$