Question

find all solutions of the equation $x^2 - 10x + 33 = 0$ by using the quadratic formula.
$x = $
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Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = -10\), \(c = 33\).

Step2: Apply quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Substitute values: \(x=\frac{-(-10)\pm\sqrt{(-10)^2 - 4\times1\times33}}{2\times1}\)
Simplify: \(x=\frac{10\pm\sqrt{100 - 132}}{2}=\frac{10\pm\sqrt{-32}}{2}\)
Simplify \(\sqrt{-32}\): \(\sqrt{-32}=\sqrt{32}\times\sqrt{-1}=4\sqrt{2}i\)
So \(x=\frac{10\pm4\sqrt{2}i}{2}=5\pm2\sqrt{2}i\)

Answer:

\(x = 5 + 2\sqrt{2}i\) or \(x = 5 - 2\sqrt{2}i\)