# Help with finding the answer?

Feb 21, 2018

$k = 3$

#### Explanation:

We need to find the value for $k$ that satisfies the equation, so we will solve for $k$.

However, first we need to integrate with respect to $x$:

${\int}_{0}^{k} \left(4 k x - 5 k\right) \mathrm{dx} = {k}^{2}$
$\implies \frac{1}{2} \cdot 4 k {x}^{2} - 5 k x {|}_{0}^{x = k} = {k}^{2}$
$\implies 2 k {x}^{2} - 5 k x {|}_{0}^{x = k} = {k}^{2}$

Now substitute in the upper and lower bounds:

$\implies 2 k {k}^{2} - 5 k k - \left(2 k \cdot {0}^{2} - 5 k \cdot 0\right) = {k}^{2}$
$\implies 2 {k}^{3} - 5 {k}^{2} = {k}^{2}$

Subtract ${k}^{2}$ from both sides:

$\implies 2 {k}^{3} - 6 {k}^{2} = 0$

A solution is in sight, we just need to factor a bit:

$\implies 2 {k}^{2} \cdot \left(k - 3\right) = 0$

We can see that for this entire thing to be zero, either $2 {k}^{2}$ must be zero, or $k - 3$ must be zero.

$2 {k}^{2} = 0$ implies $k = 0$ so there is one solution for $k$. However, since we want a non-zero answer:

$\implies k - 3 = 0$
$\implies k = 3$

Thus the answer is $k = 3$.