Question

Solve the square root equation −4x−7−−−−−−−√=2x−49−−−−−−√
−
4
𝑥
−
7
=
2
𝑥
−
49
.(1 point)
Responses

x=−7
𝑥
=
−
7
x equals negative 7

x=−21
𝑥
=
−
21
x equals negative 21

x=28
𝑥
=
28
x equals 28

x=7

Answer 1

To solve the equation

\[
-\sqrt{-4x - 7} = \sqrt{2x - 49},
\]

first, we can square both sides to eliminate the square roots. However, we need to be careful about the negative sign on the left side. Let's first rewrite it appropriately:

\[
\sqrt{-4x - 7} = -\sqrt{2x - 49}.
\]

This suggests that we will run into issues since a square root cannot equal a negative number (for real numbers). Therefore, squaring both sides won't help with this negative.

Let's square the entire equation, and keep in mind we have:
\[
\sqrt{-4x - 7} = \sqrt{49 - 2x}
\]

We can square both sides after rewriting:

\[
4x + 7 = 49 - 2x.
\]

Now, we can rearrange this to solve for \(x\):

1. Add \(2x\) to both sides:
\[
4x + 2x + 7 = 49
\]
\[
6x + 7 = 49.
\]

2. Subtract 7 from both sides:
\[
6x = 49 - 7
\]
\[
6x = 42.
\]

3. Divide both sides by 6:
\[
x = \frac{42}{6} = 7.
\]

Now we should verify our solution by substituting \(x = 7\) back into the original equation:

Left side:

\[
-\sqrt{-4(7) - 7} = -\sqrt{-28 - 7} = -\sqrt{-35},
\]

Right side:

\[
\sqrt{2(7) - 49} = \sqrt{14 - 49} = \sqrt{-35}.
\]

Both sides are equal:

\(-\sqrt{-35} = -\sqrt{-35}\).

Therefore, the solution is:

\[
\boxed{7}.
\]