Q. \(x^2+8x+16\)

Q: What is the vertex and minimum value of x^2+8x+16 ?

Vertex at x=-4. Minimum value is 16-16=0 (since it equals (x+4)^2).

Q: Solve x^2+8x+16=0 .

(x+4)^2=0 so x=-4 (double root).

Q: What are the roots and their multiplicity for x^2+8x+16 ?

Root x=-4 with multiplicity 2, because it is a perfect square.

Q: Is the expression always nonnegative?

Yes. (x+4)^2\ge 0 for all real x, and equals 0 only at x=-4.

Answer

Factor the quadratic by finding its perfect-square form.

\[
x^2+8x+16=(x+4)^2
\]

So the result is \(x+4\) squared.

Detailed Explanation

We want to simplify the expression \(x^2 + 8x + 16\). A fast way to do this is to recognize a perfect square trinomial.

Step 1: Identify the pattern for a perfect square
A perfect square has the form

\[
\left(x + a\right)^2 = x^2 + 2ax + a^2
\]

Step 2: Match coefficients with \(x^2 + 8x + 16\)
Compare

\[
x^2 + 2ax + a^2 \quad \text{with} \quad x^2 + 8x + 16
\]

This gives two equations:

\[
2a = 8
\]
\[
a^2 = 16
\]

Step 3: Solve for \(a\)
From \(2a = 8\):

\[
a = 4
\]

Check \(a^2 = 16\):

\[
4^2 = 16
\]

Step 4: Rewrite as a perfect square
Now substitute \(a = 4\) into the perfect square form:

\[
\left(x + 4\right)^2 = x^2 + 8x + 16
\]

Final Answer:

\[
x^2 + 8x + 16 = \left(x + 4\right)^2
\]

See full solution

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Algebra FAQ

What is the factoring of \(x^2+8x+16\) ?

\(x^2+8x+16=(x+4)^2\).

What is the vertex and minimum value of \(x^2+8x+16\) ?

Vertex at \(x=-4\). Minimum value is \(16-16=0\) (since it equals \((x+4)^2\)).

Solve \(x^2+8x+16=0\) .

\((x+4)^2=0\) so \(x=-4\) (double root).

Expand \((x+4)^2\) to match the expression.

\((x+4)^2=x^2+8x+16\).

What are the roots and their multiplicity for \(x^2+8x+16\) ?

Root \(x=-4\) with multiplicity \(2\), because it is a perfect square.

Is the expression always nonnegative?

Yes. \((x+4)^2\ge 0\) for all real \(x\), and equals \(0\) only at \(x=-4\).

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