Question

Use factoring to solve the equation
5xsquaredplus38xequalsminus48
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Part 1
The solution is xequals
  
enter your response here.

Answer 1

To solve the equation \( 5x^2 + 38x = -48 \) using factoring, we first rearrange the equation so that all terms are on one side:

\[
5x^2 + 38x + 48 = 0
\]

Next, we can factor the quadratic equation. To factor, we look for two numbers that multiply to \( 5 \times 48 = 240 \) and add up to \( 38 \).

After examining factors of \( 240 \), we find:

- \( 10 \) and \( 24 \) (because \( 10 + 24 = 34 \))
- \( 12 \) and \( 20 \) (because \( 12 + 20 = 32 \))
- \( 15 \) and \( 16 \) (because \( 15 + 16 = 31 \))
- \( 8 \) and \( 30 \) (because they don't add up to \( 38 \))

It seems that a better approach might be to apply the quadratic formula instead since it looks like direct factoring might be difficult.

The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a = 5 \), \( b = 38 \), and \( c = 48 \). Plugging in these values, we calculate:

1. Calculate \( b^2 - 4ac \):

\[
b^2 - 4ac = 38^2 - 4 \cdot 5 \cdot 48
\]
\[
= 1444 - 960
\]
\[
= 484
\]

2. Now we can find \( x \):

\[
x = \frac{-38 \pm \sqrt{484}}{2 \cdot 5}
\]
\[
= \frac{-38 \pm 22}{10}
\]

This gives us two solutions:

1. When using the plus sign:

\[
x = \frac{-38 + 22}{10} = \frac{-16}{10} = -1.6
\]

2. When using the minus sign:

\[
x = \frac{-38 - 22}{10} = \frac{-60}{10} = -6
\]

Thus, the two solutions to the equation \( 5x^2 + 38x + 48 = 0 \) are:

\[
x = -1.6 \quad \text{and} \quad x = -6
\]

You can enter either solution, or both if required.