Question

Solve the quadratic equation 3x2+3x+12=10
. Express its solutions in the form a±bi
.(1 point)

Answer 1

To solve the quadratic equation \(3x^2 + 3x + 12 = 10\), we first rewrite it in standard form:

\[
3x^2 + 3x + 12 - 10 = 0
\]
\[
3x^2 + 3x + 2 = 0
\]

Next, we can use the quadratic formula to solve for \(x\). The quadratic formula is given by:

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

For our equation \(3x^2 + 3x + 2 = 0\), we have:

- \(a = 3\)
- \(b = 3\)
- \(c = 2\)

Now, we can calculate the discriminant \(D\):

\[
D = b^2 - 4ac = 3^2 - 4 \cdot 3 \cdot 2 = 9 - 24 = -15
\]

Since the discriminant is negative, this means the solutions will be complex. We will now apply the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{-15}}{2 \cdot 3}
\]
\[
x = \frac{-3 \pm i\sqrt{15}}{6}
\]

We can simplify this expression:

\[
x = \frac{-3}{6} \pm \frac{i\sqrt{15}}{6}
\]
\[
x = -\frac{1}{2} \pm \frac{i\sqrt{15}}{6}
\]

Thus, the solutions in the form \(a \pm bi\) are:

\[
x = -\frac{1}{2} \pm \frac{\sqrt{15}}{6}i
\]