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Solving a Quadratic Equation by Factorization Method

If the LCM of...

Question

If the LCM of $(x^{2} + 3 x) (x^{2} + 3 x + 2)$ and $(x^{2} + 6 x + 8) (x^{2} + k x + 6)$ is $x (x + 1) {(x + 2)}^{2} (x + 3) (x + 4)$ then find $k$ .

2

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5

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Solution

The correct option is C $5$
Factor of $x^{2} + 3 x + 2 = (x + 1) (x + 2)$
Factor of $x^{2} + 6 x + 8 = (x + 2) (x + 4)$
$x (x + 3) (x + 1) (x + 2)^{2} (x + 4) (x^{2} + k x + 6) = x (x + 1) (x + 2)^{2} (x + 3) (x + 4)$
$x^{2} + k x + 6 = 1$
$x^{2} + k x + 5 = 0$
First check with the first option when $k = 2$
The quadratic equation becomes, imaginery roots.
For the second option when $k = 3$ , the equation becomes imaginery roots.
For the third option when $k = 5$
The equation becomes real and distinct roots.
$x = - 2, x = - 3$
So, the value of $k$ is $5$ .

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