Question

solve the quadratic equation.
$x^2 - 10x + 1 = 0$
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 \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = -10\), \(c = 1\).

Step2: Use quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Substitute values: \(x=\frac{10\pm\sqrt{(-10)^2 - 4(1)(1)}}{2(1)}\).

Step3: Simplify discriminant

Calculate discriminant: \(\sqrt{100 - 4}=\sqrt{96}=4\sqrt{6}\).

Step4: Simplify solutions

\(x=\frac{10\pm4\sqrt{6}}{2}=5\pm2\sqrt{6}\).

Answer:

\(5 + 2\sqrt{6}\), \(5 - 2\sqrt{6}\)