Q. \(x^2+8x+15\)
Answer
Factor the quadratic:
\[
x^2+8x+15=(x+5)(x+3)
\]
So the factored form is
\(x^2+8x+15=(x+5)(x+3)\).
Detailed Explanation
We want to simplify the quadratic expression \(x^2+8x+15\). A common first step is to factor the polynomial, if possible.
Step 1: Write the expression in factored form idea
We look for two numbers \(a\) and \(b\) such that:
- \(a\cdot b = 15\)
- \(a + b = 8\)
Step 2: Find numbers that satisfy both conditions
The factors of \(15\) are \(1\cdot 15\) and \(3\cdot 5\) (and their negative pairs). Check sums:
- \(1+15=16\) (not \(8\))
- \(3+5=8\) (yes!)
So we use \(3\) and \(5\).
Step 3: Factor the quadratic
Then the quadratic factors as:
\[
x^2+8x+15=(x+3)(x+5)
\]
Final Answer
\[
\boxed{x^2+8x+15=(x+3)(x+5)}
\]
See full solution
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Algebra FAQ
Factor \(x^2+8x+15\)?
\(x^2+8x+15=(x+3)(x+5)\).\n
Solve \(x^2+8x+15=0\)?
\( (x+3)(x+5)=0\Rightarrow x=-3,-5\).\n
Find the roots using the quadratic formula.
\(x=\frac{-8\pm\sqrt{64-60}}{2}=\frac{-8\pm2}{2}\Rightarrow x=-3,-5\).\n
Compute the axis of symmetry for \(x^2+8x+15\).
\(x=\frac{-b}{2a}=\frac{-8}{2}=-4\).\n
What is the vertex of \(y=x^2+8x+15\)?
\(y(-4)=16-32+15=-1\), so vertex is \((-4,-1)\).\n
Determine whether the quadratic is always positive.
Since it factors as \((x+3)(x+5)\) and has real zeros, it is not always positive; it’s negative between \(-5\) and \(-3\).\n
Compute the discriminant of \(x^2+8x+15\).
\(\Delta=b^2-4ac=64-60=4\).
Try solving x²+8x+15 stepwise.
Use the math tools to check.
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