Question

solve the equation by completing the square.
$x^2 + 8x + 23 = 0$

Explanation:

Step1: Move constant to right

Subtract 23 from both sides: \(x^{2}+8x=-23\)

Step2: Complete the square

Take half of 8 (\(4\)), square it (\(16\)), add to both sides: \(x^{2}+8x + 16=-23 + 16\)

Step3: Simplify both sides

Left side becomes \((x + 4)^{2}\), right side is \(-7\): \((x + 4)^{2}=-7\)

Step4: Take square root

\(x + 4=\pm\sqrt{-7}=\pm i\sqrt{7}\)

Step5: Solve for x

Subtract 4: \(x=-4\pm i\sqrt{7}\)

Answer:

\(x = -4 + i\sqrt{7}\); \(x = -4 - i\sqrt{7}\)