To solve the equation [tex]\(\sqrt{2x + 40} = \sqrt{-16 - 2x}\)[/tex], we need to isolate the variable [tex]\(x\)[/tex]. Let’s proceed step-by-step:
1. Square both sides: By squaring both sides of the equation, we can eliminate the square roots:
[tex]\[
\left(\sqrt{2x + 40}\right)^2 = \left(\sqrt{-16 - 2x}\right)^2
\][/tex]
This simplifies to:
[tex]\[
2x + 40 = -16 - 2x
\][/tex]
2. Combine like terms: To isolate [tex]\(x\)[/tex], we need to get all [tex]\(x\)[/tex] terms on one side of the equation and constants on the other side. First, add [tex]\(2x\)[/tex] to both sides:
[tex]\[
2x + 40 + 2x = -16
\][/tex]
This simplifies to:
[tex]\[
4x + 40 = -16
\][/tex]
3. Isolate [tex]\(x\)[/tex]: Subtract 40 from both sides of the equation to get the constants on one side:
[tex]\[
4x = -16 - 40
\][/tex]
Simplifying the right side, we get:
[tex]\[
4x = -56
\][/tex]
4. Solve for [tex]\(x\)[/tex]: Divide both sides by 4:
[tex]\[
x = \frac{-56}{4} = -14
\][/tex]
To ensure our solution is correct, we should verify by substituting [tex]\(x = -14\)[/tex] back into the original equation:
[tex]\[
\sqrt{2(-14) + 40} = \sqrt{-16 - 2(-14)}
\][/tex]
Checking each side:
- Left-hand side:
[tex]\[
\sqrt{2(-14) + 40} = \sqrt{-28 + 40} = \sqrt{12}
\][/tex]
- Right-hand side:
[tex]\[
\sqrt{-16 - 2(-14)} = \sqrt{-16 + 28} = \sqrt{12}
\][/tex]
Both sides simplify to [tex]\(\sqrt{12}\)[/tex], confirming that our solution [tex]\(x = -14\)[/tex] is correct.
Thus, the solution to the equation [tex]\(\sqrt{2x + 40} = \sqrt{-16 - 2x}\)[/tex] is:
[tex]\[
x = -14
\][/tex]
