Question #a01c5

Apr 15, 2017

The value of $k$ is $= 84$

Explanation:

For a perfect square, the discriminant $\Delta = 0$

$a {x}^{2} + b x + c = 0$

Here, we have

$4 {x}^{2} - 32 x - 20 + k = 0$

The discriminant is

$\Delta = {b}^{2} - 4 a c = {32}^{2} - 4 \cdot 4 \cdot \left(k - 20\right)$

${32}^{2} - 16 k + 320 = 0$

$16 k = 1344$

$k = \frac{1344}{16} = 84$

So, the equation becomes

$4 {x}^{2} - 32 x - 20 + 84 = 0$

$4 {x}^{2} - 32 x + 64 = 0$

$4 \left({x}^{2} - 8 x + 16\right) = 0$

$4 {\left(x - 4\right)}^{2} = 0$

${\left(2 \left(x - 4\right)\right)}^{2} = 0$