Question

Solve the radical equation −7x+4−−−−−−−√=1−x−−−−√+3
.(1 point)
x=

Answer 1

To solve the equation

\[
-\sqrt{-7x + 4} = \sqrt{1 - x} + 3,
\]

we first isolate the square root on one side by adding \(\sqrt{-7x + 4}\) and \(\sqrt{1 - x}\) to both sides:

\[
0 = \sqrt{1 - x} + 3 + \sqrt{-7x + 4}.
\]

Now, let's rearrange it to:

\[
\sqrt{-7x + 4} = -\sqrt{1 - x} - 3.
\]

Since the square root on the left side is non-negative, the right side must also be non-negative. Thus:

\[
-\sqrt{1 - x} - 3 \geq 0,
\]

which implies

\[
-\sqrt{1 - x} \geq 3 \quad \Rightarrow \quad \sqrt{1 - x} \leq -3,
\]

which is impossible, because square roots are always non-negative. This indicates that either:

1. There are no solutions, or
2. We made a mistake in how we isolated the terms.

Let's square both sides at some earlier point and see if we can solve it that way.

Starting from the original equation:

\[
-\sqrt{-7x + 4} = \sqrt{1 - x} + 3,
\]

we can square both sides. Square both sides gives:

\[
(-\sqrt{-7x + 4})^2 = (\sqrt{1 - x} + 3)^2.
\]

This simplifies to:

\[
-7x + 4 = (1 - x) + 6\sqrt{1 - x} + 9.
\]

Updating gives:

\[
-7x + 4 = 10 - x + 6\sqrt{1 - x}.
\]

Rearranging further,

\[
-7x + x + 4 - 10 = 6\sqrt{1 - x} \implies -6x - 6 = 6\sqrt{1 - x}.
\]

Dividing by 6 yields:

\[
-x - 1 = \sqrt{1 - x}.
\]

Again, we square both sides:

\[
(-x - 1)^2 = 1 - x.
\]

Expanding gives:

\[
x^2 + 2x + 1 = 1 - x.
\]

Bringing everything to one side gives:

\[
x^2 + 3x = 0.
\]

Factoring out \(x\):

\[
x(x + 3) = 0.
\]

Thus, we have two potential solutions:

\[
x = 0 \quad \text{or} \quad x = -3.
\]

Now, we need to check whether these solutions satisfy the original equation:

**For** \(x = 0\)

\[
-\sqrt{-7(0) + 4} = \sqrt{1 - 0} + 3 \implies -\sqrt{4} = 1 + 3 \implies -2 = 4 \quad \text{(false)}.
\]

**For** \(x = -3\)

\[
-\sqrt{-7(-3) + 4} = \sqrt{1 - (-3)} + 3 \implies -\sqrt{21 + 4} = \sqrt{4} + 3 \implies -\sqrt{25} = 2 + 3 \implies -5 = 5 \quad \text{(false)}.
\]

Since neither solution satisfies the original equation, the equation \(-\sqrt{-7x + 4} = \sqrt{1 - x} + 3\) has no solutions in real numbers.

Thus, the answer to the original equation is:

\[
\boxed{\text{No solutions}}.
\]