Two Solutions for All Quadratic Equations Quick Check

Question

4 of 54 of 5 Items
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±i5–√
1 plus or minus i Start Root 5 End Root

−1±i7–√

Answer 1

To solve the quadratic equation \(2x^2 + 4x = -12\), we first need to bring all terms to one side of the equation:

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

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

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

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

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

2. Use the quadratic formula: \[ x = \frac{-2 \pm \sqrt{-20}}{2 \cdot 1} \]

3. Simplify \(\sqrt{-20}\): \[ \sqrt{-20} = \sqrt{20} i = \sqrt{4 \cdot 5} i = 2\sqrt{5} i \]

4. Substitute back into the formula: \[ x = \frac{-2 \pm 2\sqrt{5} i}{2} \]

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

Thus, the solutions can be expressed as:

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

The correct response from the choices provided is:

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