Question

If one root of the quadratic equation ax2+bx+c=0 is double the other ,then 2b2=________

Answer 1

Hi Mate !!!


Given Equation :- ax² + bx + c = 0

It also given that the one root is double of the other.

$let \: \alpha and \: \beta \: be \: the \: roots \: of \: the \: equation \:$

$where \: \: \: \beta = 2 \alpha$

• Sum of Roots =
$\frac{ - coeff. \: of \: x}{coeff .\: of \: {x}^{2} }$

$\alpha + 2 \alpha = \frac{ - b}{a}$


$3 \alpha = \frac{ - b}{a}$


$\alpha = \frac{ - b}{3a} .....(i)$


• Product of Zeros =
$\frac{constant \: term}{coeff. \: of \: {x}^{2} }$


$\alpha \times 2 \alpha = \frac{c}{a}$


$2 { \alpha }^{2} = \frac{c}{a}$

${ \alpha }^{2} = \frac{c}{2a} ...(ii)$

Putting value of ( i ) in ( ii )

${ \alpha }^{2} = \frac{c}{2a}$

$( { \frac{ - b}{3a}) }^{2} = \frac{c}{2a}$

$\frac{ {b}^{2} }{9 {a}^{2} } = \frac{c}{2a}$


$\frac{ {b}^{2} }{9a} = \frac{c}{2}$


$2 {b}^{2} = 9ac$

Hence , 2b² = 9ac !!