Question

Find the roots of the following quadratic equation 15x - 10 root 6 x + 10 = 0. please explain properly ​

Answer 1

QuesTion :-

$\rm Find \: the \: r oots \: of \: the \: following \: quadratic \\ \rm equation \: 15 {x}^{2} - 10 \sqrt{6} x + \: 10 = 0 \: .$

Answer :-

$\to \boxed{ \rm x = \frac{ \sqrt{2} }{ \sqrt{3} } } \\ \bf or \\ \: \to \boxed{\rm \: x = \sqrt{ \frac{2}{3} } } \:$

To Find :-

→ Roots of the given quadratic equation .

Solution :-

We have a Quadratic equation -

$\to \rm \blue{ 15 {x}^{2} } - 10 \sqrt{6} x \orange{+ 10 = 0} \\ \\ \bf \red{ splitting \: the \: middle \: term \: } \\ \: \\ \to \rm \red{15 {x}^{2} - (5 \sqrt{6}} + 5 \green{\sqrt{6} )x + 10 = 0} \\ \\ \to \rm \pink{ 15 {x}^{2} - 5} \sqrt{6} \orange{ x - 5 \sqrt{6} x }+ 10 = 0 \\ \\ \to \rm \blue{ 5 (3 {x}^{2} -} \sqrt{6} x - \green{ \sqrt{6} x + 2) = 0} \\ \\ \to \rm \red{ 3 {x}^{2} - \sqrt{2} }\sqrt{3} x - \pink{ \sqrt{2} \sqrt{3} x} + 2 = 0 \\ \\ \to \rm \orange{\sqrt{3}x( \sqrt{3} x - \sqrt{2} ) -} \red{ \sqrt{2} ( \sqrt{3} x - \sqrt{2} ) = 0} \\ \\ \to \rm \green{ ( \sqrt{3} x - \sqrt{2} )}( \sqrt{3} x - \sqrt{2} ) = 0 \\ \\ \to \: {( \sqrt{3}x - \sqrt{2} ) }^{2} = 0 \\ \\ \to \rm \green{ \sqrt{3} x - } \sqrt{2} = 0 \\ \\ \to \rm \red{ \sqrt{3} x = \sqrt{2} } \\ \\ \to \boxed{ \rm x = \frac{ \sqrt{2} }{ \sqrt{3} } } \\ \bf or \\ \: \to \boxed{\rm \: x = \sqrt{ \frac{2}{3} } }$