Find the values of k for which the roots are real and equal in each of the following equations:

$x^2 - 2(5+2k)x + 3(7+10k) = 0$

Given:

Given quadratic equation is $x^2 - 2(5+2k)x + 3(7+10k) = 0$.

To do:

We have to find the values of k for which the roots are real and equal.

Solution:

Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,

$a=1, b=-2(5+2k)$ and $c=3(7+10k)$.

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

$D=[-2(5+2k)]^2-4(1)[3(7+10k)]$

$D=4(5+2k)^2-12(7+10k)$

$D=4(25+4k^2+20k)-84-120k$

$D=16k^2+80k+100-84-120k$

$D=16k^2-40k+16$

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

Therefore,

$16k^2-40k+16=0$

$8(2k^2-5k+2)=0$

$2k^2-5k+2=0$

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

$2k(k-2)-1(k-2)=0$

$(2k-1)(k-2)=0$

$2k-1=0$ or $k-2=0$

$2k=1$ or $k=2$

$k=\frac{1}{2}$ or $k=2$

The values of $k$ are $\frac{1}{2}$ and $2$. 

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Updated on: 10-Oct-2022

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