Question : If one root of the equation $Ax^{2}+Bx+C=0$ is two and a half times the others, then which of the following is true?
Option 1: $7B^{2}=3CA$
Option 2: $7B^{2}=4CA$
Option 3: $7B^{2}=36CA$
Option 4: $10B^{2}=49CA$
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Correct Answer: $10B^{2}=49CA$
Solution :
$Ax^{2} + Bx + C = 0$
Let one root be $p$ and then the other root will be $2.5p$.
$⇒ p+2.5p=-\frac{B}{A}$ (sum of the roots)
$⇒ 3.5p=-\frac{B}{A}$......(i)
Also, $p\times2.5p=\frac{C}{A}$ (product of the roots)
$⇒ 2.5p^{2}=\frac{C}{A}$........(ii)
From equation (i) and (ii)
$\frac{2.5 \times B^{2}}{(3.5\times3.5\times A^{2})} =\frac{C}{A}$
$⇒ B^{2} = 4.9CA$
$⇒10B^{2} = 49CA$
Hence, the correct answer is $10B^{2} = 49CA$.
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