Question

Use the quadratic formula to solve 2x2−3x−8=0


Round your answer to the nearest hundredth if necessary.

Reference Sheet - Quadratic Formula

(2 points)
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Answer 1

To solve the quadratic equation \(2x^2 - 3x - 8 = 0\) using the quadratic formula, we'll start with the formula itself:

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

In this case, the coefficients are:
- \(a = 2\)
- \(b = -3\)
- \(c = -8\)

Now, let's substitute these values into the quadratic formula:

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

\[
b^2 = (-3)^2 = 9
\]
\[
4ac = 4 \cdot 2 \cdot (-8) = -64
\]
\[
b^2 - 4ac = 9 - (-64) = 9 + 64 = 73
\]

2. Now, use the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{73}}{2 \cdot 2}
\]
\[
x = \frac{3 \pm \sqrt{73}}{4}
\]

Next, calculate the two possible solutions:

1. Calculate \(x_1\):

\[
x_1 = \frac{3 + \sqrt{73}}{4}
\]

2. Calculate \(x_2\):

\[
x_2 = \frac{3 - \sqrt{73}}{4}
\]

Now we can find the approximate values:

- First evaluate \(\sqrt{73} \approx 8.544\).

Now substitute back into the equations:

1. \(x_1\):

\[
x_1 \approx \frac{3 + 8.544}{4} \approx \frac{11.544}{4} \approx 2.886
\]

2. \(x_2\):

\[
x_2 \approx \frac{3 - 8.544}{4} \approx \frac{-5.544}{4} \approx -1.386
\]

Finally, rounding to the nearest hundredth, the solutions are:

\[
x \approx 2.89
\]
\[
x \approx -1.39
\]

Thus, the final answers are:

\[
x = 2.89
\]
\[
x = -1.39
\]