Question

Solve −2x2+3x−1=−2−2x2+3x−1=−2 using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.(1 point)

Answer 1

To solve the quadratic equation \(-2x^2 + 3x - 1 = -2\), we first rearrange it into the standard form \(ax^2 + bx + c = 0\).

Starting with the original equation:

\[
-2x^2 + 3x - 1 = -2
\]

we can add 2 to both sides to get:

\[
-2x^2 + 3x - 1 + 2 = 0
\]

which simplifies to:

\[
-2x^2 + 3x + 1 = 0
\]

Next, we multiply the entire equation by -1 to simplify it further:

\[
2x^2 - 3x - 1 = 0
\]

Now, we identify \(a = 2\), \(b = -3\), and \(c = -1\). We will use the quadratic formula, which is given by:

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

First, we calculate the discriminant:

\[
b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot (-1) = 9 + 8 = 17
\]

Now, we substitute \(a\), \(b\), and the discriminant into the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{17}}{2(2)} = \frac{3 \pm \sqrt{17}}{4}
\]

Now we compute the two potential solutions:

1. For the positive solution:

\[
x_1 = \frac{3 + \sqrt{17}}{4}
\]

2. For the negative solution:

\[
x_2 = \frac{3 - \sqrt{17}}{4}
\]

Now we approximate \(\sqrt{17} \approx 4.123\). Using this approximation:

1. Calculate \(x_1\):

\[
x_1 \approx \frac{3 + 4.123}{4} = \frac{7.123}{4} \approx 1.78075 \text{ (rounds to } 1.78 \text{)}
\]

2. Calculate \(x_2\):

\[
x_2 \approx \frac{3 - 4.123}{4} = \frac{-1.123}{4} \approx -0.28075 \text{ (rounds to } -0.28 \text{)}
\]

Thus, the solutions rounded to the nearest hundredths are:

**Smaller solution**: \(-0.28\)
**Larger solution**: \(1.78\)

Final answer:

\[
\boxed{-0.28} \text{ and } \boxed{1.78}
\]