Question

Solve the quadratic equation 2x2+4x=−12
. Which of the following expresses its solutions in the form a±bi
?(1 point)
Responses

−2±2i5–√
negative 2 plus or minus 2 i Start Root 5 End Root

−1±i5–√
negative 1 plus or minus i Start Root 5 End Root

−1±i7–√
negative 1 plus or minus i Start Root 7 End Root

1±i5–√

Answer 1

To solve the quadratic equation \(2x^2 + 4x = -12\), we first rearrange it into standard form:

\[
2x^2 + 4x + 12 = 0
\]

Next, we can simplify the equation by dividing all terms by 2:

\[
x^2 + 2x + 6 = 0
\]

Now we will use the quadratic formula to find the solutions:

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

In our equation, \(a = 1\), \(b = 2\), and \(c = 6\). Plugging these values into the quadratic formula gives:

\[
b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 6 = 4 - 24 = -20
\]

Since the discriminant is negative, we will have complex solutions.

Now, substituting back into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{-20}}{2 \cdot 1}
\]

We can rewrite \(\sqrt{-20}\) as \(\sqrt{20}i\):

\[
\sqrt{-20} = \sqrt{20}i = \sqrt{4 \cdot 5}i = 2\sqrt{5}i
\]

So, substituting this back into the formula:

\[
x = \frac{-2 \pm 2\sqrt{5}i}{2}
\]

Simplifying this yields:

\[
x = -1 \pm \sqrt{5}i
\]

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

\[
-1 \pm i\sqrt{5}
\]

So the correct answer from the provided options is:

**−1±i5–√ (negative 1 plus or minus i Start Root 5 End Root)**