To express the quadratic equation \(3x^2 + 18x + 24 = 0\) in terms of the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we need to identify the coefficients \(a\), \(b\), and \(c\).
From the equation \(3x^2 + 18x + 24 = 0\):
- \(a = 3\)
- \(b = 18\)
- \(c = 24\)
Now, we'll substitute these values into the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ x = \frac{-18 \pm \sqrt{18^2 - 4 \cdot 3 \cdot 24}}{2 \cdot 3} \]
Calculating \(b^2 - 4ac\):
\[ b^2 = 18^2 = 324 \] \[ 4ac = 4 \cdot 3 \cdot 24 = 288 \] \[ b^2 - 4ac = 324 - 288 = 36 \]
Now substituting back into the formula:
\[ x = \frac{-18 \pm \sqrt{36}}{6} \]
Calculating the square root:
\[ \sqrt{36} = 6 \]
Now we can write the final expression:
\[ x = \frac{-18 \pm 6}{6} \]
This simplifies to two solutions:
- \(x = \frac{-18 + 6}{6} = \frac{-12}{6} = -2\)
- \(x = \frac{-18 - 6}{6} = \frac{-24}{6} = -4\)
Thus, the quadratic formula form for the equation \(3x^2 + 18x + 24 = 0\) is:
\[ x = \frac{-18 \pm 6}{6} \]
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