Q. \( (x-6)^2 \)

Q: Expand (x-6)^2.

(x-6)^2 = (x-6)(x-6)=x^2-12x+36.

Q: What is the vertex and minimum value of (x-6)^2?

Rewrite as (x-6)^2 , so the vertex is (6,0). The minimum value is 0 at x=6.

Q: Solve (x-6)^2=0.

(x-6)^2=0 implies x-6=0, so x=6.

Q: Solve (x-6)^2=9.

(x-6)^2=9 \Rightarrow x-6=\pm 3. So x=9 or x=3.

Answer

\( (x-6)^2 \) is a perfect square. Use \( (a-b)^2=a^2-2ab+b^2 \) with \(a=x\) and \(b=6\).

\[
\begin{aligned}
(x-6)^2 &= x^2 – 2(x)(6) + 6^2 \\
&= x^2 – 12x + 36
\end{aligned}
\]

Detailed Explanation

We want to expand the expression \( (x-6)^2 \).

Step 1: Recognize the square of a binomial.
For any binomial \( (a-b)^2 \), the identity is:

\[
(a-b)^2 = a^2 – 2ab + b^2
\]

Here, match \(x-6\) to \(a-b\). We have:

\(a = x\) and \(b = 6\).

Step 2: Substitute into the identity.
Substitute \(a = x\) and \(b = 6\) into

\[
(a-b)^2 = a^2 – 2ab + b^2
\]

to get:

\[
(x-6)^2 = x^2 – 2(x)(6) + 6^2
\]

Step 3: Simplify each term.

First term:

\[
x^2 = x^2
\]

Middle term:

\[
-2(x)(6) = -12x
\]

Last term:

\[
6^2 = 36
\]

Step 4: Combine the results.
Now add the simplified terms:

\[
(x-6)^2 = x^2 – 12x + 36
\]

Final Answer:
\[
(x-6)^2 = x^2 – 12x + 36
\]

See full solution

Graph

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Algebra FAQ

Expand \((x-6)^2\).

\((x-6)^2 = (x-6)(x-6)=x^2-12x+36\).

What is the vertex and minimum value of \((x-6)^2\)?

Rewrite as \( (x-6)^2 \), so the vertex is \((6,0)\). The minimum value is \(0\) at \(x=6\).

Solve \((x-6)^2=0\).

\((x-6)^2=0\) implies \(x-6=0\), so \(x=6\).

Solve \((x-6)^2=9\).

\((x-6)^2=9 \Rightarrow x-6=\pm 3\). So \(x=9\) or \(x=3\).

Find \(\sqrt{(x-6)^2}\).

\(\sqrt{(x-6)^2}=|x-6|\).

What is the derivative of \((x-6)^2\)?

\(\frac{d}{dx}(x-6)^2=2(x-6)\).

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