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 Root1±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–√

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 Root1±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–√

GPT-4o mini · Answer

To solve the quadratic equation $ 2x^2 + 4x = -12 $, we will first rearrange it into standard form. 1. Move all terms to one side of the equation: $$ 2x^2 + 4x + 12 = 0 $$ 2. Next, we can simplify the equation by dividing all terms by 2: $$ x^2 + 2x + 6 = 0 $$ 3. Now we will use the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1 $, $ b = 2 $, and $ c = 6 $. 4. First, we calculate the discriminant $ b^2 - 4ac $: $$ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 6 = 4 - 24 = -20 $$ 5. Since the discriminant is negative, the solutions will involve complex numbers. Now substituting into the quadratic formula: $$ x = \frac{-2 \pm \sqrt{-20}}{2 \cdot 1} = \frac{-2 \pm \sqrt{20} i}{2} $$ 6. Simplifying further: $$ x = \frac{-2}{2} \pm \frac{\sqrt{20} i}{2} = -1 \pm \frac{\sqrt{20}}{2} i $$ 7. Since $ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} $: $$ x = -1 \pm \frac{2\sqrt{5}}{2} i = -1 \pm \sqrt{5} i $$ Thus, the solutions can be expressed as: $$ x = -1 \pm i\sqrt{5} $$ From the provided responses, the correct answer is: $$ -1 \pm i\sqrt{5} $$