Q. \(\)

Q: How do you factor x^2-64?

Use difference of squares: x^2-64=x^2-8^2=(x-8)(x+8).

Q: How do you solve x^2-64=16?

Rewrite: x^2-64=16 \Rightarrow x^2=80 \Rightarrow x=\pm \sqrt{80}=\pm 4\sqrt{5}.

Q: What is the domain and range of y=x^2-64?

Domain: all real numbers. Range: y\ge -64, with minimum at x=0.

Q: What are the roots of x^2-64 and their multiplicities?

Roots are x=8 and x=-8. Each has multiplicity 1 since factors are (x-8)(x+8).

Q: How do you find the vertex of y=x^2-64?

For y=x^2-64, vertex form is already given: vertex at (0,-64).

Answer

We solve the equation \(x^2-64=0\).

Rewrite \(64\) as a square: \(x^2-8^2=0\).

Factor using the difference of squares: \(x^2-8^2=(x-8)(x+8)=0\).

So \(x-8=0\) or \(x+8=0\), giving \(x=8\) or \(x=-8\).

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

Detailed Explanation

We are solving the expression \(x^2 – 64\). To “solve” it in a typical algebra way, we factor it completely (this makes the structure clear and gives the solution to the implied equation \(x^2-64=0\)).

Step 1: Recognize a difference of squares.

The expression \(x^2 – 64\) is a difference of squares because \(64 = 8^2\). Rewrite \(64\) as \(8^2\):

\[
x^2 – 64 = x^2 – 8^2
\]

Step 2: Use the difference of squares formula.

The identity is

\[
a^2 – b^2 = (a-b)(a+b)
\]

Here, \(a = x\) and \(b = 8\). Substitute into the formula:

\[
x^2 – 8^2 = (x-8)(x+8)
\]

Step 3: Write the final factored form.

\[
x^2 – 64 = (x-8)(x+8)
\]

Step 4: If the intention is to solve \(x^2 – 64 = 0\), find the values of \(x\).

Set the factored form equal to zero:

\[
(x-8)(x+8) = 0
\]

Then use the zero product property: if a product is zero, at least one factor is zero.

\[
x-8 = 0 \quad \text{or} \quad x+8 = 0
\]

Step 5: Solve each simple equation.

First:

\[
x-8 = 0 \Rightarrow x = 8
\]

Second:

\[
x+8 = 0 \Rightarrow x = -8
\]

Final Answer:

\[
x^2 – 64 = (x-8)(x+8)
\]

If solving \(x^2 – 64 = 0\), then \(x = 8\) or \(x = -8\).

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Algebra FAQ

What are the solutions to \(x^2-64=0\)?

Solve \(x^2=64\), so \(x=\pm 8\).

How do you factor \(x^2-64\)?

Use difference of squares: \(x^2-64=x^2-8^2=(x-8)(x+8)\).

How do you solve \(x^2-64=16\)?

Rewrite: \(x^2-64=16 \Rightarrow x^2=80 \Rightarrow x=\pm \sqrt{80}=\pm 4\sqrt{5}\).

What is the domain and range of \(y=x^2-64\)?

Domain: all real numbers. Range: \(y\ge -64\), with minimum at \(x=0\).

What are the roots of \(x^2-64\) and their multiplicities?

Roots are \(x=8\) and \(x=-8\). Each has multiplicity \(1\) since factors are \((x-8)(x+8)\).

How do you find the vertex of \(y=x^2-64\)?

For \(y=x^2-64\), vertex form is already given: vertex at \((0,-64)\).

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