Question

what are the solutions to the quadratic equation $x^2 + 4x + 8 = 0$. use the keypad to enter the answer as a single simplified expression in the box. positive solution: $x = \square$ negative solution: $x = \square$

Explanation:

Step1: Recall quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step2: Identify coefficients

Here $a=1$, $b=4$, $c=8$

Step3: Compute discriminant

$\Delta = b^2-4ac = 4^2-4(1)(8)=16-32=-16$

Step4: Simplify square root of discriminant

$\sqrt{-16} = \sqrt{16}\times\sqrt{-1}=4i$

Step5: Substitute into quadratic formula

$x=\frac{-4\pm4i}{2}=-2\pm2i$

Step6: Separate positive/negative imaginary solutions

The solution with the positive imaginary part is $-2+2i$, and the one with the negative imaginary part is $-2-2i$.

Answer:

Positive solution: $x = -2+2i$
Negative solution: $x = -2-2i$