If $x^{2} + 7 x + 1 = 0$ and $(a^{2} - 4 a) x + b x - 3 = 0$ have both the roots in common, then find the value of $a$ .

$1$

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$4$

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$2$

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$3$

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Solution

The correct option is D
$3$

If two equations have the same roots or represent the same equation, then the ratio of the corresponding coefficients is same.
$x^{2} + 7 x + 1 = 0$ and $(a^{2} - 4 a) x + b x - 3 = 0$ represent the same equation.

$\Rightarrow$ We can compare the coefficients.

$\Rightarrow \frac{a^{2} - 4 a}{1} = \frac{b}{7} = \frac{- 3}{1}$

$\Rightarrow a^{2} - 4 a = - 3$

$\Rightarrow a^{2} - 4 a + 3 = 0$

$\Rightarrow (a - 3) (a - 1) = 0$

$\Rightarrow a = 1$ or $a = 3$

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