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±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

−1±i5–√

Answer 1

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

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

Next, we simplify this equation by dividing all terms by 2:

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

Now, we will use the quadratic formula, which is given by:

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

where \(a = 1\), \(b = 2\), and \(c = 6\).

First, we calculate the discriminant (\(b^2 - 4ac\)):

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

Since the discriminant is negative, this indicates that the solutions will be complex numbers. We can now plug in the values into the quadratic formula:

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

We can simplify this further:

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

Since \(\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}\), we have:

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

Now, we can express the solutions as:

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

Thus, the correct answer is:

\(-1 \pm i\sqrt{5}\).