Q. \(<\) \(x^2-36\) \(>\)

Q: Factorize x^2-36 and give its roots.

x^2-36=(x-6)(x+6). Roots: x=6 and x=-6.

Q: Solve x^2-36=0.

Set x^2-36=0\Rightarrow x^2=36\Rightarrow x=\pm 6.

Q: Expand (x-6)(x+6) to verify it equals x^2-36.

(x-6)(x+6)=x^2+6x-6x-36=x^2-36.

Q: Find the y-intercept of y=x^2-36.

Substitute x=0: y=0-36=-36. So the intercept is -36.

Q: Determine where y=x^2-36 is zero.

Solve x^2-36=0\Rightarrow x=\pm 6. So zeros are at x=-6 and x=6.

Answer

To factor \(x^2 – 36\), recognize it as a difference of squares:

\[
x^2 – 36 = x^2 – 6^2 = (x-6)(x+6)
\]

Final result: \((x-6)(x+6)\)

Detailed Explanation

We want to simplify the expression \(x^2 – 36\).

Step 1: Recognize a factoring pattern.

Notice that \(x^2 – 36\) is a difference of squares because it has the form

\[a^2 – b^2 = (a-b)(a+b).\]

Step 2: Identify \(a\) and \(b\).

We match \(x^2 – 36\) to \(a^2 – b^2\).

\[a^2 = x^2 \quad \Rightarrow \quad a = x,\]

\[b^2 = 36 \quad \Rightarrow \quad b = 6.\]

Step 3: Apply the difference of squares formula.

Substitute \(a=x\) and \(b=6\) into \((a-b)(a+b)\):

\[x^2 – 36 = (x-6)(x+6).\]

Final Answer:

\[x^2 – 36 = (x-6)(x+6).\]

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Algebra FAQ

Factorize \(x^2-36\) and give its roots.

\(x^2-36=(x-6)(x+6)\). Roots: \(x=6\) and \(x=-6\).

Solve \(x^2-36=0\).

Set \(x^2-36=0\Rightarrow x^2=36\Rightarrow x=\pm 6\).

Use the difference of squares to factor \(x^2-36\).

\(x^2-36=x^2-6^2=(x-6)(x+6)\).

Expand \((x-6)(x+6)\) to verify it equals \(x^2-36\).

\((x-6)(x+6)=x^2+6x-6x-36=x^2-36\).

Find the \(y\)-intercept of \(y=x^2-36\).

Substitute \(x=0\): \(y=0-36=-36\). So the intercept is \(-36\).

Determine where \(y=x^2-36\) is zero.

Solve \(x^2-36=0\Rightarrow x=\pm 6\). So zeros are at \(x=-6\) and \(x=6\).

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