Question

solve the quadratic equation.
$x^2 - 3x - 13 = 0$
write one exact solution in each box. you can add or remove boxes. if there are no real solutions, remove all boxes.

Explanation:

Step1: Identify coefficients

For \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = -3\), \(c = -13\).

Step2: Use quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Substitute values: \(x=\frac{-(-3)\pm\sqrt{(-3)^2 - 4(1)(-13)}}{2(1)}\)

Step3: Simplify discriminant

Calculate \(b^2 - 4ac = 9 + 52 = 61\)

Step4: Solve for x

\(x=\frac{3\pm\sqrt{61}}{2}\)

Answer:

\(\frac{3 + \sqrt{61}}{2}\), \(\frac{3 - \sqrt{61}}{2}\)