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:
- Calculate \( b^2 - 4ac \):
\[ b^2 - 4ac = 38^2 - 4 \cdot 5 \cdot 48 \] \[ = 1444 - 960 \] \[ = 484 \]
- Now we can find \( x \):
\[ x = \frac{-38 \pm \sqrt{484}}{2 \cdot 5} \] \[ = \frac{-38 \pm 22}{10} \]
This gives us two solutions:
- When using the plus sign:
\[ x = \frac{-38 + 22}{10} = \frac{-16}{10} = -1.6 \]
- 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.
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