Question

Solve the radical equation sqrt
(−
7
x
+
4)
= sqrt
(16
−
x)
−7x+4
​
=
16−x
​
.

x
=
x=

Answer 1

To solve the equation

\[
\sqrt{-7x + 4} = \sqrt{16 - x},
\]

we will first square both sides to eliminate the square roots. This gives us:

\[
-7x + 4 = 16 - x.
\]

Next, we will rearrange the equation to isolate \(x\). We can add \(7x\) to both sides:

\[
4 = 16 - x + 7x,
\]

which simplifies to:

\[
4 = 16 + 6x.
\]

Next, subtract 16 from both sides:

\[
4 - 16 = 6x,
\]

which simplifies to:

\[
-12 = 6x.
\]

Now, divide both sides by 6:

\[
x = -2.
\]

Now that we have a potential solution \(x = -2\), we must check it in the original equation to ensure it does not result in any extraneous solutions.

Substituting \(x = -2\) back into the original equation:

\[
\sqrt{-7(-2) + 4} = \sqrt{16 - (-2)}.
\]

Calculating both sides:

Left side:

\[
\sqrt{14 + 4} = \sqrt{18} = 3\sqrt{2}.
\]

Right side:

\[
\sqrt{16 + 2} = \sqrt{18} = 3\sqrt{2}.
\]

Since both sides are equal, \(x = -2\) is indeed a valid solution.

Thus, the solution to the equation is

\[
\boxed{-2}.
\]