Question

Solve the equation 2, x, squared, minus, 18, x, plus, 15, equals, 82x
2
−18x+15=8 to the nearest tenth.

Answer 1

To solve the equation \(2x^2 - 18x + 15 = 82\), we start by rearranging it into standard form. We can do this by moving all terms to one side of the equation.

1. Subtract 82 from both sides:
\[
2x^2 - 18x + 15 - 82 = 0
\]
This simplifies to:
\[
2x^2 - 18x - 67 = 0
\]

2. Next, we can use the quadratic formula to solve for \(x\). The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our equation \(2x^2 - 18x - 67 = 0\), the coefficients are:
- \(a = 2\)
- \(b = -18\)
- \(c = -67\)

3. Calculate the discriminant (\(b^2 - 4ac\)):
\[
b^2 - 4ac = (-18)^2 - 4(2)(-67) = 324 + 536 = 860
\]

4. Now we can substitute the values into the quadratic formula:
\[
x = \frac{-(-18) \pm \sqrt{860}}{2 \cdot 2}
\]
Simplifying:
\[
x = \frac{18 \pm \sqrt{860}}{4}
\]

5. Next, let's calculate \(\sqrt{860}\):
\[
\sqrt{860} \approx 29.32
\]

6. Substitute \(\sqrt{860}\) back into the formula:
\[
x = \frac{18 \pm 29.32}{4}
\]

7. Now we calculate the two possible values for \(x\):
- First solution:
\[
x_1 = \frac{18 + 29.32}{4} = \frac{47.32}{4} \approx 11.83
\]
- Second solution:
\[
x_2 = \frac{18 - 29.32}{4} = \frac{-11.32}{4} \approx -2.83
\]

8. Finally, rounding the positive solution to the nearest tenth:
\[
x_1 \approx 11.8
\]

Therefore, the solution to the equation \(2x^2 - 18x + 15 = 82\) to the nearest tenth is approximately \(\boxed{11.8}\).