Question

or
Solve the quadratic equation.
–
3x2+2x+2=0
Write one exact solution in each box. You can add or remove boxes. If there are no real solutions, remove all boxes.

,

Answer 1

To solve the quadratic equation \(-3x^2 + 2x + 2 = 0\), we can use the quadratic formula:

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

where \(a = -3\), \(b = 2\), and \(c = 2\).

First, we calculate the discriminant:

\[
b^2 - 4ac = 2^2 - 4(-3)(2) = 4 + 24 = 28
\]

Since the discriminant is positive, there are two real solutions.

Now, substituting the values into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{28}}{2(-3)}
\]

Simplifying \(\sqrt{28}\):

\[
\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}
\]

Now substitute back into the formula:

\[
x = \frac{-2 \pm 2\sqrt{7}}{-6}
\]

Breaking it into two separate parts:

1. For the positive solution:
\[
x_1 = \frac{-2 + 2\sqrt{7}}{-6} = \frac{2\sqrt{7} - 2}{6} = \frac{\sqrt{7} - 1}{3}
\]

2. For the negative solution:
\[
x_2 = \frac{-2 - 2\sqrt{7}}{-6} = \frac{-2 - 2\sqrt{7}}{-6} = \frac{2 + 2\sqrt{7}}{6} = \frac{1 + \sqrt{7}}{3}
\]

Thus, the two exact solutions are:

1. \(x_1 = \frac{\sqrt{7} - 1}{3}\)
2. \(x_2 = \frac{1 + \sqrt{7}}{3}\)

You can fill in the boxes with these two solutions. If you want to differentiate them in two separate boxes, it would look like:

- Box 1: \(\frac{\sqrt{7} - 1}{3}\)
- Box 2: \(\frac{1 + \sqrt{7}}{3}\)