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

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

−1±i7–√

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 \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 2 \), and \( c = 6 \).

Calculating the discriminant \( b^2 - 4ac \):

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

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

Now we can plug in the values into the quadratic formula:

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

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

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

Substituting back into the equation gives:

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

Thus, the solutions can be expressed in the form \( -1 \pm i\sqrt{5} \).

The correct response is:

\(-1 \pm i\sqrt{5}\) (which can be written as \( -1 \pm i\sqrt{5} \)).