Question

2x² + 2x + 5 = 0
\\(\frac{-1 pm isqrt{11}}{2}\\) \\(\frac{-1 pm 3i}{2}\\) \\(\frac{2 pm sqrt{40}}{4}\\) \\(\frac{-1 pm isqrt{6}}{2}\\)

Explanation:

Step1: Identify quadratic coefficients

For $2x^2 + 2x + 5 = 0$, $a=2$, $b=2$, $c=5$

Step2: Use quadratic formula

Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step3: Calculate discriminant

$\Delta = b^2 - 4ac = 2^2 - 4(2)(5) = 4 - 40 = -36$

Step4: Simplify square root of discriminant

$\sqrt{-36} = \sqrt{36 \times (-1)} = 6i$

Step5: Substitute into formula

$x = \frac{-2 \pm 6i}{2 \times 2} = \frac{-2 \pm 6i}{4}$

Step6: Simplify the fraction

$x = \frac{-1 \pm 3i}{2}$ (correction: $\sqrt{-36}=6i$, $\frac{-2\pm6i}{4}=\frac{-1\pm3i}{2}$ – correction to discriminant calculation: $b^2-4ac=4-40=-36$, $\sqrt{-36}=6i$, so $\frac{-2\pm6i}{4}=\frac{-1\pm3i}{2}$)

Wait, correction to step 3: $4ac=4*2*5=40$, $b^2=4$, so $4-40=-36$, $\sqrt{-36}=6i$, so $\frac{-2\pm6i}{4}=\frac{-1\pm3i}{2}$. The correct answer is $\frac{-1 \pm 3i}{2}$

Corrected Answer:

$\frac{-1 \pm 3i}{2}$

Step1: Identify $a,b,c$

$a=2$, $b=2$, $c=5$

Step2: Apply quadratic formula

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

Step3: Compute discriminant

$\Delta=2^2-4(2)(5)=4-40=-36$

Step4: Simplify root of discriminant

$\sqrt{-36}=6i$

Step5: Substitute values

$x=\frac{-2\pm6i}{4}$

Step6: Reduce the fraction

$x=\frac{-1\pm3i}{2}$

Answer:

$\frac{-1 \pm i\sqrt{11}}{2}$