Q. \(x^2 = 64\)

Q: What is the general solution to x^2=64?

x=\pm 8, since squaring gives two values whose squares equal 64.

Q: How do you solve x^2=64 using square roots?

Take square roots: x=\sqrt{64} or x=-\sqrt{64}. Thus x=8 or x=-8.

Q: Why are there two solutions here?

Because x^2 is the same for x and -x. Both 8^2 and (-8)^2 equal 64.

Q: Solve x^2=64 if x is restricted to be nonnegative.

With x\ge 0, only x=8 works. The solution x=-8 is excluded.

Q: Solve x^2=64 if x is restricted to be negative.

With x<0, only x=-8 works. The solution x=8 is excluded.

Q: What is the solution to x^2=81 as a related example?

x=\pm 9, because 81 is a perfect square: 9^2=81 and (-9)^2=81.

Answer

We solve \(x^2=64\) by taking square roots:

\[
x=\pm \sqrt{64}=\pm 8
\]

Final result: \(x=8\) or \(x=-8\).

Detailed Explanation

Step 1: Rewrite the equation in a solvable form.

You are given:

\[x^2 = 64\]

Step 2: Take the square root of both sides.

The square root undoes the squaring, but because \(x^2\) was squared, both positive and negative solutions are possible. So we write:

\[x = \pm \sqrt{64}\]

Step 3: Compute the square root of \(64\).

\[64 = 8^2 \Rightarrow \sqrt{64} = 8\]

Step 4: Substitute \(8\) into the solution.

\[x = \pm 8\]

Final Answer:

\[x = 8 \quad \text{or} \quad x = -8\]

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Algebra FAQ

What is the general solution to \(x^2=64\)?

\(x=\pm 8\), since squaring gives two values whose squares equal \(64\).

How do you solve \(x^2=64\) using square roots?

Take square roots: \(x=\sqrt{64}\) or \(x=-\sqrt{64}\). Thus \(x=8\) or \(x=-8\).

Why are there two solutions here?

Because \(x^2\) is the same for \(x\) and \(-x\). Both \(8^2\) and \((-8)^2\) equal \(64\).

Solve \(x^2=64\) if \(x\) is restricted to be nonnegative.

With \(x\ge 0\), only \(x=8\) works. The solution \(x=-8\) is excluded.

Solve \(x^2=64\) if \(x\) is restricted to be negative.

With \(x<0\), only \(x=-8\) works. The solution \(x=8\) is excluded.

What is the solution to \(x^2=81\) as a related example?

\(x=\pm 9\), because \(81\) is a perfect square: \(9^2=81\) and \((-9)^2=81\).

We solve x^2=64 step by step.
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