Question

28. Find the roots by the method of parfect square : (x + 2) (x + 3) - 240 = 0​

Answer 1

Question :-

$</p><p> \rm find \: the \: roots \: by \: the \: method \: of \: perfect \\ \rm \: square : (x + 2)(x + 3) - 240 = 0$

Answer :-

$\to \boxed{ \rm x = \pm \bigg \{ \frac{ \sqrt{961} - 5 }{2} \bigg \}} \:$

Used Concept :-

→ Perfect square method for find roots -

• Step 1 -Divide all terms by the coefficient of x² (a)

• Step 2 -Move the term (c/a) to the right side of the equation.

• Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

Solution :-

We have ,

$\to \large \rm (x + 2)(x + 3) - 240 = 0 \\ \\ \sf \: simplify \: it \\ \\\large \to \rm {x}^{2} + 3x + 2x + 6 - 240 = 0 \\ \\ \laarge \to \rm {x}^{2} + 5x - 234 = 0 \\ \\ \star \large \bold{ \underline{ step \: (1) }}\: \\ \to \large \rm here \: coefficient \: of \: {x}^{2} \: is \: 1 \: \: \\ \rm \large \: \: \: \: so \: no \: effect \: of \: divided \: by \: 1 \\ \\ \large \mapsto \rm \: {x}^{2} + 5x - 234 = 0 \\ \\ \star \: \bold{ \underline{ step \: (2) }}\: \\ \\ \large \to \rm move \: the \: constant \: term \: to \: \\ \rm \large \: \: \: \: right \: side \\ \\ \to \rm {x}^{2} + 5x = 234 \\ \\ \star \: \bold{ \underline{ step \: (3) }}\: \\ \\ \to \large \rm complete \: square \: by \: adding \: something \\ \: \\ \large \to \rm {x}^{2} + 5x + { \bigg (\frac{5}{2} \bigg) }^{2} - { \bigg( \frac{5}{2} \bigg) }^{2} = 234 \\ \\ \large \to \rm { \bigg(x + \frac{5}{2} \bigg) }^{2} = 234 + \frac{25}{4} \\ \\ \to \rm \large { \bigg(x + \frac{5}{2} \bigg)}^{2} = \frac{936 + 25}{4} \\ \\ \large \to \rm{ \bigg(x + \frac{5}{2} \bigg)}^{2} \: = \frac{961}{4} \\ \\ \rm \large taking \: \sqrt{} \: on \: both \: sides \\ \\ \to \rm x + \frac{5}{2} = \frac{ \sqrt{961} }{2} \\ \\ \to \boxed{ \rm x = \pm \bigg \{ \frac{ \sqrt{961} - 5 }{2} \bigg \}}$