To solve the equation \(3x^2 - 8x - 23 = -4x - 9\), let's first rearrange all terms to one side of the equation.
Starting with the original equation:
\[ 3x^2 - 8x - 23 + 4x + 9 = 0 \]
Combine like terms:
\[ 3x^2 - 4x - 14 = 0 \]
Now, we will use the quadratic formula to solve for \(x\):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 3\), \(b = -4\), and \(c = -14\).
First, calculate the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4(3)(-14) = 16 + 168 = 184 \]
Now, apply the quadratic formula:
\[ x = \frac{-(-4) \pm \sqrt{184}}{2 \cdot 3} = \frac{4 \pm \sqrt{184}}{6} \]
Next, simplify \(\sqrt{184}\):
\[ \sqrt{184} = \sqrt{16 \cdot 11} = 4\sqrt{11} \]
Now substitute back into the formula:
\[ x = \frac{4 \pm 4\sqrt{11}}{6} = \frac{2 \pm 2\sqrt{11}}{3} \]
This gives us two expressions for \(x\):
\[ x_1 = \frac{2 + 2\sqrt{11}}{3} \]
\[ x_2 = \frac{2 - 2\sqrt{11}}{3} \]
Now, we need to compute these values to the nearest tenth. First, let's evaluate \(2\sqrt{11}\):
\[ \sqrt{11} \approx 3.3166 \quad \Rightarrow \quad 2\sqrt{11} \approx 6.6332 \]
Now calculate the two values:
- For \(x_1\):
\[ x_1 \approx \frac{2 + 6.6332}{3} = \frac{8.6332}{3} \approx 2.8777 \quad \Rightarrow \quad x_1 \approx 2.9 \text{ (to the nearest tenth)} \]
- For \(x_2\):
\[ x_2 \approx \frac{2 - 6.6332}{3} = \frac{-4.6332}{3} \approx -1.5444 \quad \Rightarrow \quad x_2 \approx -1.5 \text{ (to the nearest tenth)} \]
Thus, the solutions to the equation \(3x^2 - 8x - 23 = -4x - 9\) are:
\[ x \approx 2.9 \quad \text{and} \quad x \approx -1.5 \]