Q. \(x^2 = 8\)

Q: Solve x^2=8.

Take square roots: x=\pm\sqrt{8}=\pm 2\sqrt{2}.

Q: Why are there two solutions to x^2=8?

Squaring removes the sign, so both x=2\sqrt{2} and x=-2\sqrt{2} satisfy the equation.

Q: How do you simplify \sqrt{8}?

\sqrt{8}=\sqrt{4\cdot 2}=2\sqrt{2}.

Q: What’s the solution using a radical equation step-by-step?

From x^2=8, write x=\sqrt{8} or x=-\sqrt{8}, giving x=\pm 2\sqrt{2}.

Q: Check your solutions: do they satisfy x^2=8?

(2\sqrt{2})^2=4\cdot 2=8 and (-2\sqrt{2})^2=8.

Q: Solve x^2=8 over the real numbers only.

Real solutions are x=\pm 2\sqrt{2} since 8 is positive.

Q: If the equation were x^2=-8, what would change?

No real solutions; over complex numbers, x=\pm 2\sqrt{2}\,i.

Answer

We solve the equation \(x^2=8\) by taking square roots of both sides:

\[
x=\pm\sqrt{8}=\pm\sqrt{4\cdot2}=\pm 2\sqrt{2}
\]

Final answer: \(x=2\sqrt{2}\) or \(x=-2\sqrt{2}\).

Detailed Explanation

We are asked to solve the equation \(x^2 = 8\).

Step 1: Understand the equation.
The equation \(x^2 = 8\) means that \(x\) squared equals 8.

Step 2: Take the square root of both sides.
To undo the squaring, take the square root of both sides:

\[
x = \sqrt{8}
\]

Step 3: Remember both square roots.
Because squaring removes the sign, \(x^2 = 8\) has two solutions: one positive and one negative. So we write:

\[
x = \pm \sqrt{8}
\]

Step 4: Simplify \(\sqrt{8}\).
We simplify \(\sqrt{8}\) by factoring inside the square root:

\[
\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4}\sqrt{2} = 2\sqrt{2}
\]

Step 5: Substitute back.
So the solutions are:

\[
x = \pm 2\sqrt{2}
\]

Final Answer.
\[
x = 2\sqrt{2} \quad \text{or} \quad x = -2\sqrt{2}
\]

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Algebra FAQ

Solve \(x^2=8\).

Take square roots: \(x=\pm\sqrt{8}=\pm 2\sqrt{2}\).

Why are there two solutions to \(x^2=8\)?

Squaring removes the sign, so both \(x=2\sqrt{2}\) and \(x=-2\sqrt{2}\) satisfy the equation.

How do you simplify \(\sqrt{8}\)?

\(\sqrt{8}=\sqrt{4\cdot 2}=2\sqrt{2}\).

What’s the solution using a radical equation step-by-step?

From \(x^2=8\), write \(x=\sqrt{8}\) or \(x=-\sqrt{8}\), giving \(x=\pm 2\sqrt{2}\).

Check your solutions: do they satisfy \(x^2=8\)?

\((2\sqrt{2})^2=4\cdot 2=8\) and \((-2\sqrt{2})^2=8\).

Solve \(x^2=8\) over the real numbers only.

Real solutions are \(x=\pm 2\sqrt{2}\) since 8 is positive.

If the equation were \(x^2=-8\), what would change?

No real solutions; over complex numbers, \(x=\pm 2\sqrt{2}\,i\).

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