Question
or
Solve the quadratic equation.
–
3x^2+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.
,
Answers
GPT-4o mini
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 \(c = 2\).
1. Calculate the discriminant (\(b^2 - 4ac\)):
\[
b^2 - 4ac = 2^2 - 4(-3)(2) = 4 + 24 = 28
\]
2. Since the discriminant is positive, there are two real solutions. Now, we can compute the solutions using the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{28}}{2(-3)} = \frac{-2 \pm \sqrt{28}}{-6}
\]
3. Simplifying \(\sqrt{28}\):
\[
\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}
\]
4. Substitute back into the formula:
\[
x = \frac{-2 \pm 2\sqrt{7}}{-6} = \frac{2 \mp 2\sqrt{7}}{6} = \frac{1 \mp \sqrt{7}}{3}
\]
Thus, the two exact solutions are:
\[
x_1 = \frac{1 + \sqrt{7}}{3}
\]
\[
x_2 = \frac{1 - \sqrt{7}}{3}
\]
You can place one of the solutions in each box:
\(\frac{1 + \sqrt{7}}{3}\)
\(\frac{1 - \sqrt{7}}{3}\)
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = -3\), \(b = 2\), and \(c = 2\).
1. Calculate the discriminant (\(b^2 - 4ac\)):
\[
b^2 - 4ac = 2^2 - 4(-3)(2) = 4 + 24 = 28
\]
2. Since the discriminant is positive, there are two real solutions. Now, we can compute the solutions using the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{28}}{2(-3)} = \frac{-2 \pm \sqrt{28}}{-6}
\]
3. Simplifying \(\sqrt{28}\):
\[
\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}
\]
4. Substitute back into the formula:
\[
x = \frac{-2 \pm 2\sqrt{7}}{-6} = \frac{2 \mp 2\sqrt{7}}{6} = \frac{1 \mp \sqrt{7}}{3}
\]
Thus, the two exact solutions are:
\[
x_1 = \frac{1 + \sqrt{7}}{3}
\]
\[
x_2 = \frac{1 - \sqrt{7}}{3}
\]
You can place one of the solutions in each box:
\(\frac{1 + \sqrt{7}}{3}\)
\(\frac{1 - \sqrt{7}}{3}\)