Question

solve the quadratic equation.
\\(x^2 + 4x + 29 = 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 \(x^2 + 4x + 29 = 0\), \(a = 1\), \(b = 4\), \(c = 29\).

Step2: Use quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Substitute values: \(x=\frac{-4\pm\sqrt{4^2 - 4(1)(29)}}{2(1)}\)

Step3: Simplify discriminant

Calculate \(b^2 - 4ac\): \(16 - 116=-100\)
So \(x=\frac{-4\pm\sqrt{-100}}{2}\)

Step4: Simplify square root

\(\sqrt{-100}=\sqrt{100}\times\sqrt{-1}=10i\)
Thus \(x=\frac{-4\pm10i}{2}\)

Step5: Simplify fraction

Divide numerator terms by 2: \(x=-2\pm5i\)

Answer:

\(-2 + 5i\), \(-2 - 5i\)