Question

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

1±i5–√
1 plus or minus 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

−2±2i5–√
negative 2 plus or minus 2 i Start Root 5 End Root
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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 this equation by dividing all terms by 2:

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

Now, we will use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 2\), and \(c = 6\).

Calculating the discriminant:

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

Since the discriminant is negative, the solutions will involve complex numbers.

Now applying the quadratic formula:

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

This simplifies to:

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

We can simplify \(\sqrt{20}\):

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

So we substitute this back into our formula:

\[
x = \frac{-2 \pm 2\sqrt{5}i}{2} = -1 \pm \sqrt{5}i
\]

Therefore, the solutions can be expressed as:

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

The correct answer is:

\(-1 \pm i\sqrt{5}\) which corresponds to the second option in your responses.