Question

Solve the equation by factoring.
5x^2 +8x -69=0
x=____
(Simplify your answer. Use a comma to separate answers as needed)

Answer 1

To solve the equation \( 5x^2 + 8x - 69 = 0 \) by factoring, we first need to find two numbers that multiply to \( a \cdot c = 5 \cdot (-69) = -345 \) and add to \( b = 8 \).

We are looking for two numbers that multiply to \(-345\) and add to \(8\). After considering various factor pairs, we find:

- The numbers are \( 15 \) and \(-23\) because \( 15 \cdot (-23) = -345 \) and \( 15 + (-23) = -8\).

Next, we can rewrite the middle term using these numbers:

\[
5x^2 + 15x - 23x - 69 = 0
\]

Now, we can factor by grouping:

\[
(5x^2 + 15x) + (-23x - 69) = 0
\]
\[
5x(x + 3) - 23(x + 3) = 0
\]

Now factor out the common term \((x + 3)\):

\[
(5x - 23)(x + 3) = 0
\]

Now we can set each factor equal to zero:

1. \(5x - 23 = 0\)
2. \(x + 3 = 0\)

Solving for \(x\) in the first equation:

\[
5x = 23
\]
\[
x = \frac{23}{5}
\]

Solving for \(x\) in the second equation:

\[
x = -3
\]

The solutions are:

\[
x = \frac{23}{5}, -3
\]

Thus, the final answers are:

\[
\boxed{\frac{23}{5}, -3}
\]