Q. \(x^2 – 81\)

Q: Prove x^2-81 factors over integers.

Use difference of squares: \,x^2-81=x^2-9^2=(x-9)(x+9)\,.

Q: Solve x^2-81=0.

Set (x-9)(x+9)=0, so x-9=0 or x+9=0. Solutions: x=9 or x=-9.

Q: Factor x^2-81 completely.

The complete factorization is (x-9)(x+9). No further factoring over integers since both are linear.

Q: Find the roots of the equation x^2-81=0 and their multiplicities.

Roots are x=9 and x=-9. Each appears once, so both have multiplicity 1.

Q: Rewrite x^2-81 as a product using a formula.

Since 81=9^2, apply a^2-b^2=(a-b)(a+b): x^2-81=(x-9)(x+9).

Q: Determine whether x^2-81 is positive, negative, or zero.

Factor: (x-9)(x+9). It is zero at x=\pm 9, positive for |x|>9, and negative for -9<x<9.

Answer

We solve the equation \(x^2-81=0\).

Rewrite \(81\) as \(9^2\):

\[
x^2-9^2=0
\]
\[
(x-9)(x+9)=0
\]

So \(x-9=0\) or \(x+9=0\), giving \(x=9\) or \(x=-9\).

Final result: \(x=\pm 9\).

Detailed Explanation

We want to factor the expression \(x^2 – 81\).

Step 1: Recognize a difference of squares.

The expression \(x^2 – 81\) matches the pattern

\[
a^2 – b^2 = (a-b)(a+b).
\]

Here, we identify \(a^2 = x^2\) so \(a = x\), and \(b^2 = 81\) so \(b = 9\) (since \(9^2 = 81\)).

Step 2: Substitute into the difference of squares formula.

Using \(a = x\) and \(b = 9\), we get

\[
x^2 – 81 = (x-9)(x+9).
\]

Final Answer:

\[
x^2 – 81 = (x-9)(x+9).
\]

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Algebra FAQ

Prove \(x^2-81\) factors over integers.

Use difference of squares: \(\,x^2-81=x^2-9^2=(x-9)(x+9)\,.\)

Solve \(x^2-81=0\).

Set \((x-9)(x+9)=0\), so \(x-9=0\) or \(x+9=0\). Solutions: \(x=9\) or \(x=-9\).

Factor \(x^2-81\) completely.

The complete factorization is \((x-9)(x+9)\). No further factoring over integers since both are linear.

Find the roots of the equation \(x^2-81=0\) and their multiplicities.

Roots are \(x=9\) and \(x=-9\). Each appears once, so both have multiplicity \(1\).

Rewrite \(x^2-81\) as a product using a formula.

Since \(81=9^2\), apply \(a^2-b^2=(a-b)(a+b)\): \(x^2-81=(x-9)(x+9)\).

Determine whether \(x^2-81\) is positive, negative, or zero.

Factor: \((x-9)(x+9)\). It is zero at \(x=\pm 9\), positive for \(|x|>9\), and negative for \(-9<x<9\).

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