Q. \(x^2 + 8x + 15 = 0\)

Q: Solve x^2+8x+15=0 using the quadratic formula.

x=\frac{-8\pm\sqrt{64-60}}{2}=\frac{-8\pm2}{2}, giving x=-3 or x=-5.

Q: What is the discriminant of x^2+8x+15=0?

D=b^2-4ac=8^2-4(1)(15)=64-60=4.

Q: Do these roots come out real and distinct?

Yes, since D=4>0, there are two distinct real solutions.

Q: How can you complete the square for x^2+8x+15=0?

x^2+8x+15=(x+4)^2-1=0, so (x+4)^2=1 and x=-3,-5.

Q: Check the solutions by substitution.

For x=-3: 9-24+15=0. For x=-5: 25-40+15=0.

Answer

We solve \(x^2+8x+15=0\) by factoring.

Find two numbers that multiply to \(15\) and add to \(8\): \(3\) and \(5\).

\[
x^2+8x+15=(x+3)(x+5)=0
\]

\[
x=-3 \quad \text{or} \quad x=-5
\]

Final result: \(x=-3\) or \(x=-5\).

Detailed Explanation

We want to solve the quadratic equation

\[
x^2 + 8x + 15 = 0
\]

Step 1: Factor the quadratic.

To factor \(x^2 + 8x + 15\), we look for two numbers that:

1. Multiply to \(15\).
2. Add to \(8\).

The numbers \(3\) and \(5\) work because:

\[
3 \cdot 5 = 15
\]
\[
3 + 5 = 8
\]

So the quadratic factors as:

\[
x^2 + 8x + 15 = (x + 3)(x + 5)
\]

Step 2: Set each factor equal to zero.

Because

\[
(x + 3)(x + 5) = 0
\]

we must have either:

\[
x + 3 = 0
\]
or
\[
x + 5 = 0
\]

Step 3: Solve each linear equation.

First:

\[
x + 3 = 0
\]
\[
x = -3
\]

Second:

\[
x + 5 = 0
\]
\[
x = -5
\]

Final Answer:

\[
x = -3 \text{ or } x = -5
\]

See full solution

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Algebra FAQ

Solve \(x^2+8x+15=0\) using factoring.

\((x+3)(x+5)=0\), so \(x=-3\) or \(x=-5\).

Solve \(x^2+8x+15=0\) using the quadratic formula.

\(x=\frac{-8\pm\sqrt{64-60}}{2}=\frac{-8\pm2}{2}\), giving \(x=-3\) or \(x=-5\).

What is the discriminant of \(x^2+8x+15=0\)?

\(D=b^2-4ac=8^2-4(1)(15)=64-60=4\).

Do these roots come out real and distinct?

Yes, since \(D=4>0\), there are two distinct real solutions.

How can you complete the square for \(x^2+8x+15=0\)?

\(x^2+8x+15=(x+4)^2-1=0\), so \((x+4)^2=1\) and \(x=-3,-5\).

Check the solutions by substitution.

For \(x=-3\): \(9-24+15=0\). For \(x=-5\): \(25-40+15=0\).

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