Find all real numbers $k$ such that
\[\left\| k \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} \right\| = 2 \sqrt{13}.\]Enter all the solutions, separated by commas.
We have that
\begin{align*}
\left\| k \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} \right\| &= \left\| \begin{pmatrix} 2k \\ -3k \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} \right\| \\
&= \left\| \begin{pmatrix} 2k - 4 \\ -3k - 7 \end{pmatrix} \right\| \\
&= \sqrt{(2k - 4)^2 + (-3k - 7)^2} \\
&= 13k^2 + 26k + 65,
\end{align*}so we want to solve the equation $\sqrt{13k^2 + 26k + 65} = 2 \sqrt{13}$.  Squaring both sides, we get $13k^2 + 26k + 65 = 52$, which simplifies to
\[13k^2 + 26k + 13 = 13(k + 1)^2 = 0.\]The only solution $k = \boxed{-1}.$
-1
Level 2
Precalculus
Problem: Find all real numbers $k$ such that
\[\left\| k \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} \right\| = 2 \sqrt{13}.\]Enter all the solutions, separated by commas.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the problem, we start by rewriting the expression inside the norm. We have:

k * (2, -3) - (4, 7) = (2k - 4, -3k - 7).

Next, we need to compute the norm of this vector:

|| (2k - 4, -3k - 7) || = sqrt((2k - 4)^2 + (-3k - 7)^2).

Setting this equal to 2√13 gives us the equation:

sqrt((2k - 4)^2 + (-3k - 7)^2) = 2√13.

Squaring both sides eliminates the square root:

(2k - 4)^2 + (-3k - 7)^2 = (2√13)^2.

Calculating the right side:

(2√13)^2 = 4 * 13 = 52.

Now we expand the left side:

(2k - 4)^2 = 4k^2 - 16k + 16,
(-3k - 7)^2 = 9k^2 + 42k + 49.

Combining these gives:

4k^2 - 16k + 16 + 9k^2 + 42k + 49 = 52.

This simplifies to:

13k^2 + 26k + 65 = 52.

Subtracting 52 from both sides results in:

13k^2 + 26k + 13 = 0.

Dividing the entire equation by 13 simplifies it to:

k^2 + 2k + 1 = 0.

Factoring this quadratic equation yields:

(k + 1)^2 = 0.

Thus, the only solution is:

k + 1 = 0, which gives k = -1.

Therefore, the final answer is -1.<|END_OF_LATENT|> The final answer is $-1$.