If $2$ is a root of the quadratic equation $3x^2 + px - 8 = 0$ and the quadratic equation $4x^2 - 2px + k = 0$ has equal roots, find the value of k.

Given:

$2$ is a root of the quadratic equation $3x^2 + px -8 = 0$ and the quadratic equation $4x^2 -2px + k = 0$ has equal roots.

To do:

We have to find the value of k.

Solution:

If $m$ is a root of the quadratic equation $ax^2+bx+c=0$ then it satisfies the given equation.

Therefore,

$3x^2 + px -8 = 0$

$3(2)^2 + p(2) -8 = 0$

$3(4)+2p-8=0$

$12+2p-8=0$

$4=2p$

$p=\frac{4}{2}$

$p=2$

Substituting the value of $p$ in $4x^2 -2px + k = 0$, we get,

$4x^2 - 2(2)x + k = 0$

$4x^2-4x+k=0$

Comparing the quadratic equation  $4x^2-4x+k=0$ with the standard form of a quadratic equation $ax^2+bx+c=0$,

$a=4, b=-4$ and $c=k$

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is $D=b^2-4ac$.

$D=(-4)^2-4(4)(k)$

$D=16-16k$

The given quadratic equation has equal roots if $D=0$.

Therefore,

$16-16k=0$

$16=16k$

$k=\frac{16}{16}$

$k=1$

The value of $k$ is $1$.

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Updated on: 10-Oct-2022

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