Question

Solve the equation 3, x, squared, minus, 8, x, minus, 23, equals, minus, 4, x, minus, 93x
2
−8x−23=−4x−9 to the nearest tenth.

Answer 1

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:

1) For \(x_1\):

\[
x_1 \approx \frac{2 + 6.6332}{3} = \frac{8.6332}{3} \approx 2.8777 \quad \Rightarrow \quad x_1 \approx 2.9 \text{ (to the nearest tenth)}
\]

2) 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:

\[
x \approx 2.9 \quad \text{and} \quad x \approx -1.5
\]