To solve the quadratic equation [tex]\(2x^2 + 16x + 34 = 0\)[/tex], we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a = 2\)[/tex], [tex]\(b = 16\)[/tex], and [tex]\(c = 34\)[/tex].
1. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = 16^2 - 4 \cdot 2 \cdot 34 \][/tex]
[tex]\[ \Delta = 256 - 272 \][/tex]
[tex]\[ \Delta = -16 \][/tex]
2. Compute the solutions using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
[tex]\[ x = \frac{-16 \pm \sqrt{-16}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{-16 \pm 4i}{4} \][/tex]
Simplifying, we get:
[tex]\[ x = \frac{-16}{4} \pm \frac{4i}{4} \][/tex]
[tex]\[ x = -4 \pm i \][/tex]
So, the solutions for the equation [tex]\(2x^2 + 16x + 34 = 0\)[/tex] are:
[tex]\[ x = -4 + i \][/tex]
[tex]\[ x = -4 - i \][/tex]
Among the provided choices, the correct one is:
[tex]\[ \boxed{-4 \pm i} \][/tex]
