If 2 is a root of the equation x2 + ax + 12 = 0 and the quadratic
Question:

If 2 is a root of the equation $x^{2}+a x+12=0$ and the quadratic equation $x^{2}+a x+q=0$ has equal roots, then $q=$

(a) 12
(b) 8
(c) 20
(d) 16

Solution:

$x=2$ is the common roots given quadric equation are $x^{2}+a x+12=0$, and $x^{2}+a x+q=0$

Then find the value of q.

Here, $x^{2}+a x+12=0$……(1)

$x^{2}+a x+q=0 \ldots .(2)$

Putting the value of $x=2$ in equation (1) we get

$2^{2}+a \times 2+12=0$

$4+2 a+12=0$

$2 a=-16$

$a=-8$

Now, putting the value of $a=-8$ in equation ( 2 ) we get

$x^{2}-8 x+q=0$

Then,

$a_{2}=1, b_{2}=-8$ and,$c_{2}=q$

As we know that $D_{1}=b^{2}-4 a c$

Putting the value of $a_{2}=1, b_{2}=-8$ and, $c_{2}=q$

$=(-8)^{2}-4 \times 1 \times q$

$=64-4 q$

The given equation will have equal roots, if $D=0$

$64-4 q=0$

$4 q=64$

$q=\frac{64}{4}$

$q=16$

Thus, the correct answer is (d)

 

 

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