Solve the following quadratic equations for x and give your answer correct to 2 decimal places :

(i) x² - 5x - 10 = 0

(ii) x² + 7x = 7

(i) The given equation is x² - 5x - 10 = 0

Comparing it with ax² + bx + c = 0, we get
a = 1 , b = -5 , c = -10

By using formula, $x = \dfrac{-b ± \sqrt{b^2 - 4ac}}{2a}$

we obtain:

$\Rightarrow x = \dfrac{-(-5) ± \sqrt{(-5)^2 - 4\times 1 \times -10}}{2 \times 1} \\[1em] \Rightarrow x = \dfrac{5 ± \sqrt{25 + 40}}{2} \\[1em] \Rightarrow x = \dfrac{5 ± \sqrt{65}}{2} \\[1em] \Rightarrow x = \dfrac{5 + \sqrt{65}}{2} \text{ or } \dfrac{5 - \sqrt{65}}{2} \\[1em] \text{ Also } \sqrt{65} = 8.062 (\text{From tables}) \\[1em] \Rightarrow x = \dfrac{5 + 8.062}{2} \text { or } \dfrac{5 - 8.062}{2} \\[1em] \Rightarrow x = \dfrac{13.062}{2} \text{ or } \dfrac{-3.062}{2} \\[1em] \Rightarrow x = 6.531 \text{ or } -1.531 \\[1em] x = 6.53 \text{ or } -1.53 \text{ (correct to two decimal places) }$

Hence roots of the given equations are 6.53 , -1.53.

(ii) Given, x² + 7x = 7

or , x² + 7x - 7 = 0

The given equation is x² + 7x - 7 = 0

Comparing it with ax² + bx + c = 0, we get
a = 1 , b = 7 , c = -7

By using formula, $x = \dfrac{-b ± \sqrt{b^2 - 4ac}}{2a}$

we obtain:

$\Rightarrow x = \dfrac{-(7) ± \sqrt{(7)^2 - 4\times 1 \times -7}}{2 \times 1} \\[1em] \Rightarrow x = \dfrac{-7 ± \sqrt{49 + 28}}{2} \\[1em] \Rightarrow x = \dfrac{-7 ± \sqrt{77}}{2} \\[1em] \Rightarrow x = \dfrac{-7 + \sqrt{77}}{2} \text{ or } \dfrac{-7 - \sqrt{77}}{2} \\[1em] \text{ Also } \sqrt{77} = 8.775 (\text{From tables}) \\[1em] \Rightarrow x = \dfrac{-7 + 8.775}{2} \text { or } \dfrac{-7 - 8.775}{2} \\[1em] \Rightarrow x = \dfrac{1.775}{2} \text{ or } \dfrac{-15.775}{2} \\[1em] \Rightarrow x = 0.885 \text{ or } -7.885 \\[1em] \Rightarrow x = 0.89\text{ or } -7.89 \text{ (correct to two decimal places) }$

Hence roots of the given equations are 0.89 , -7.89.