Question

Find the roots of the following quadratic equation:-
$\sf \sqrt{3x {}^{2} } + 10x + 7 \sqrt{3} = 0$
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Answer 1

$\sf \bf \huge {\boxed {\mathbb {QUESTION}}}$

$\tt 1: \:Find\: the \:roots\: of \:the\: following\: quadratic \\\tt equation :$

$\tt \sqrt{{3x}^{2}} + 10x + 7 \sqrt{3} = 0$

$\sf \bf \huge {\boxed {\mathbb {SOLUTION}}}$

${\pink {\underline {\bf {\pmb {The\:roots \:of \:the\:quadratic \:equation (Factorisation \:method)}}}}}$

$\bf \sqrt{3}{x}^{2}\:+\: 10x \:+\: 7\sqrt{3} = 0$

$\:\:\:\:\:\:\:\:\:\:\:\:\leadsto\sf 21{x}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \swarrow \searrow \\\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\sf + 3x \: \: + 7x$

$\bf \implies \sqrt{3}{x}^{2} +3x+7x+7\sqrt{3} = 0$

$\bf \implies \sqrt{3}x\Big(x+\sqrt{3}\Big)+7\Big(x+\sqrt{3}\Big)=0\longrightarrow \big(\because 3=\sqrt{3}\times \sqrt{3}\big)$

$\bf \implies \Big(x+\sqrt{3}\Big) \Big(\sqrt{3}x+7\Big)=0$

$\bf \implies x+\sqrt{3}=0\:\:or\:\:\sqrt{3}x+7=0$

$\bf \implies x=-\sqrt{3} \:\:or\:\:sqrt{3}x=-7$

$\implies{\blue {\boxed {\boxed {\purple{\mathfrak {x=-\sqrt{3} \:\:or\:\:x=\dfrac{-7}{\sqrt{3}}}}}}}}$

${\underbrace {\red {\underline {\red {\overline {\red {\pmb {\sf {{\therefore}} The\:roots \:of \:the\:quadratic \:equation \:\sqrt{3}{x}^{2} +3x+7x+7\sqrt{3} = 0\:are\:\sqrt{3}\:or\:\dfrac{-7}{\sqrt{3}}}}}}}}}}$

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$\sf \bf \huge {\boxed {\mathbb {EXTRA\:INFORMATION}}}$

${\green{\underline {\pmb{Types\: of\: methods\: to\: solve\: quadratic \:equation}}}}$

$\sf Factoring\:method$

$\sf Completing\: the\: Square\:method$

$\sf Quadratic\: Formula\:method$

$\sf Grapical \:method$

Answer 2

$\boxed {\boxed{ { \green{ \bold{ \underline{Verified \: Answer \: }}}}}}$

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$\bigstar{\underline{\underline{{\sf\ \red{ SOLUTION-}}}}}$

√3x2 + 10x + 7√3 = 0

⇒ √3x2 + 3x + 7x + 7√3 = 0

⇒ √3x( x + √3) + 7(x + √3) = 0

⇒ √3x + 7 = 0 or x + √3 = 0

x = - 7√3 or - √3 are two roots of equation.

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