Question

3. Solve the following quadratic equations by
factorisation method :
$6 + \sqrt{3 {x}^{2} } + 7x = 3$
​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

$\bf \: Solve \: the \: following \: by \: factorisation \: method :$

$\rm :\longmapsto\: 6 + \sqrt{ {3x}^{2} } + 7x = 3$

$\large\underline{\sf{Solution-}}$

Consider,

$\rm :\longmapsto\: 6 + \sqrt{ {3x}^{2} } + 7x = 3$

$\rm :\longmapsto\:\sqrt{ {3x}^{2} } = 3 - 6 - 7x$

$\rm :\longmapsto\:\sqrt{ {3x}^{2} } = - \: 3 \: - \: 7x$

$\rm :\longmapsto\:\sqrt{ {3x}^{2} } = - \:( 3 \: + \: 7x)$

On squaring both sides, we get

$\rm :\longmapsto\:3 {x}^{2} = 9 + {49x}^{2} + 42x$

$\: \: \: \: \: \: \: \: \: \boxed{ \sf \: \because \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}$

$\rm :\longmapsto\: {46x}^{2} + 42x + 9 = 0$

Divide both sides by 46, we get

$\rm :\longmapsto\: {x}^{2} + \dfrac{42}{46} x + \dfrac{9}{46} = 0$

$\rm :\longmapsto\: {x}^{2} + \dfrac{(21 + 21)}{46} x + \dfrac{9}{46} = 0$

$\rm :\longmapsto\: {x}^{2} + \dfrac{(21 + 21 + 3 \sqrt{3} - 3 \sqrt{3} )}{46} x + \dfrac{9}{46} = 0$

$\rm :\longmapsto\: {x}^{2} + \dfrac{(21 + 3 \sqrt{3})+(21 - 3 \sqrt{3})}{46} x + \dfrac{9}{46} = 0$

$\rm :\longmapsto\: {x}^{2} + \dfrac{(21 + 3 \sqrt{3}) }{46}x + \dfrac{(21 - 3 \sqrt{3}) }{46}x + \dfrac{9}{46} = 0$

$\rm \:x\bigg( x + \dfrac{(21 + 3 \sqrt{3}) }{46}\bigg) + \dfrac{(21 - 3 \sqrt{3}) }{46}\bigg(x + \dfrac{9}{46} \times \dfrac{46}{21 - 3 \sqrt{3}} \bigg) = 0$

$\bigg(\red{ \sf \: \frac{9 \times 46}{21 + 3 \sqrt{3}} \times \frac{21 -3 \sqrt{3} }{21 - 3 \sqrt{3}} = \frac{414(21 - 3 \sqrt{3}) }{441 - 27} = \frac{414(21 - 3 \sqrt{3}) }{414} = 21 - 3 \sqrt{3}} \bigg)$

$\rm \:x\bigg( x + \dfrac{(21 + 3 \sqrt{3}) }{46}\bigg) + \dfrac{(21 - 3 \sqrt{3}) }{46}\bigg(x + \dfrac{21 + 3 \sqrt{3} }{46} \bigg) = 0$

$\rm :\longmapsto\:{\bigg(x + \dfrac{(21 + 3 \sqrt{3}) }{46}\bigg) }{\bigg(x + \dfrac{(21 - 3 \sqrt{3}) }{46} \bigg) } = 0$

$\rm :\implies\:x = - \dfrac{(21 + 3 \sqrt{3}) }{46} \: \: or \: - \: \dfrac{(21 - 3 \sqrt{3}) }{46}$

Basic Concept Used :-

Splitting of middle terms :-

• In order to factorize,  ax² + bx + c we have to find numbers p and q such that p + q = b and pq = ac.

• After finding p and q, we split the middle term in the quadratic as px + qx and get desired factors by grouping the terms.