Question

solve the quadratic equation.
\\( x^2 + 8x + 52 = 0 \\)
write one exact, simplified solution in each box. you can add or remove boxes. if a solution is not a real number, write it in the form \\( a + bi \\) or \\( a - bi \\), where \\( a \\) and \\( b \\) are real numbers.

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = 8\), \(c = 52\).

Step2: Use quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Calculate discriminant \(\Delta=b^2 - 4ac=(8)^2-4\times1\times52 = 64 - 208=-144\)

Step3: Substitute into formula

\(x=\frac{-8\pm\sqrt{-144}}{2\times1}=\frac{-8\pm12i}{2}\)
Simplify: \(x=-4\pm6i\)

Answer:

\(-4 + 6i\), \(-4 - 6i\)