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–√
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–√
1 plus or minus i Start Root 5 End Root

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

Answer 1

To solve the quadratic equation \( 2x^2 + 4x = -12 \), we first rewrite it in the standard form \( ax^2 + bx + c = 0 \).

Starting with the equation:

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

Now we can simplify this equation by dividing every term by 2:

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

Next, we can use the quadratic formula:

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

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

\[
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}
\]

Calculating the discriminant:

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

Since the discriminant is negative, we have complex solutions. Now substituting back into the quadratic formula:

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

To simplify \( \sqrt{-20} \):

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

Now substituting this back into the equation:

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

This simplifies to:

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

Thus, the solutions are:

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

The correct choice from the given options is:

**−1±i5–√ (negative 1 plus or minus \( i \sqrt{5} \))**