The equation formed by multiplying each root of $a x^{2} + b x + c = 0$ by 2 is $x^{2} + 36 x + 24 = 0$ .Which one of the following is correct ?

b c = a^{2}

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b c = 36 a^{2}

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b c = 72 a^{2}

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b c = 108 a^{2}

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Solution

The correct option is D $b c = 108 a^{2}$
let $p, q$ be roots of equation $a x^{2} + b x + c = 0$

So $p + q = (- \frac{b}{a})$ and $p q = c / a$

$\Rightarrow b = - a (p + q), c = a p q$

New equation is $x^{2} + 36 x + 24 = 0$ and roots are $2 p, 2 q$

So $2 p + 2 q = - 36$

$\Rightarrow p + q = - 18$

$2 p \times 2 q = 24$

$\Rightarrow p q = 6$

Then, value of $b c$ is $[- a (p + q)] [a p q] = - a (- 18) a 6 = 108 a^{2}$

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The equation formed by multiplying each root of ax2+bx+c=0 by 2 is x2+36x+24=0.Which one of the following is correct ?

The correct option is D bc=108a2let p,q be roots of equation ax2+bx+c=0So p+q=(−ba) and pq=c/a⇒b=−a(p+q),c=apqNew equation is x2+36x+24=0 and roots are 2p,2qSo 2p+2q=−36⇒p+q=−182p×2q=24⇒pq=6Then, value of bc is [−a(p+q)][apq]=−a(−18)a6=108a2

The equation formed by multiplying each root of $a x^{2} + b x + c = 0$ by 2 is $x^{2} + 36 x + 24 = 0$ .Which one of the following is correct ?