Question

#14
$x^2 + 8x + 16$
$a\cdot c = 1\cdot 8$
...
$a = 1$
$b = 8$
$c = 16$
...
$(x + 2)(x + $

Explanation:

Step1: Identify the quadratic form

We have the quadratic expression \(x^{2}+8x + 16\). For a quadratic of the form \(ax^{2}+bx + c\), here \(a = 1\), \(b=8\), \(c = 16\).

Step2: Factor the quadratic

We need two numbers that multiply to \(a\times c=1\times16 = 16\) and add up to \(b = 8\). The numbers are \(4\) and \(4\) (since \(4\times4=16\) and \(4 + 4=8\)). Wait, but in the given work, there was a miscalculation earlier. The correct factoring:
The quadratic \(x^{2}+8x + 16\) is a perfect square trinomial, which factors as \((x + 4)(x + 4)=(x + 4)^{2}\). But if we follow the factoring by splitting the middle term (even though the earlier split was wrong), let's correct it. We need two numbers \(m\) and \(n\) such that \(m\times n=16\) and \(m + n=8\). So \(m = 4\), \(n = 4\). Then \(x^{2}+8x + 16=x^{2}+4x+4x + 16=x(x + 4)+4(x + 4)=(x + 4)(x + 4)=(x + 4)^{2}\). But if we consider the incorrect split in the image (with \(2\) and \(6\) maybe? No, \(2\times6 = 12
eq16\)). Wait, the correct factors for \(x^{2}+8x + 16\) are \((x + 4)(x + 4)\). But if we assume the intended split was wrong and we correct it:
Let's do it properly. For \(x^{2}+8x + 16\), since \(a = 1\), \(b = 8\), \(c = 16\). The formula for factoring \(ax^{2}+bx + c\) when \(a = 1\) is to find two numbers that multiply to \(c\) and add to \(b\). So numbers are \(4\) and \(4\). So \(x^{2}+8x + 16=(x + 4)(x + 4)\).

But if we take the work in the image where they had \(2\) and some other number, maybe a typo. But the correct factoring of \(x^{2}+8x + 16\) is \((x + 4)^{2}\) or \((x + 4)(x + 4)\).

Wait, maybe the original problem was a different quadratic? Wait, the given quadratic is \(x^{2}+8x + 16\). Let's confirm:

\(x^{2}+8x + 16\):

• \(a = 1\), \(b = 8\), \(c = 16\)
• Discriminant \(D=b^{2}-4ac=64 - 64=0\), so it's a perfect square.

So the correct factoring is \((x + 4)(x + 4)\) or \((x + 4)^{2}\).

But if we follow the factoring by splitting the middle term correctly:

\(x^{2}+8x + 16=x^{2}+4x+4x + 16\)

\(=x(x + 4)+4(x + 4)\)

\(=(x + 4)(x + 4)\)

Answer:

The correct factored form of \(x^{2}+8x + 16\) is \(\boldsymbol{(x + 4)^{2}}\) (or \((x + 4)(x + 4)\)).