Question

find all solutions of the equation $x^2 - 6x + 59 = 0$ by using the quadratic formula.
$x = \square$
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Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Here, \(a = 1\), \(b=-6\), \(c = 59\).

Step2: Calculate the Discriminant

First, find the discriminant \(D=b^2 - 4ac\).
Substitute \(a = 1\), \(b=-6\), \(c = 59\) into the discriminant formula:
\(D=(-6)^2-4\times1\times59=36 - 236=-200\)

Step3: Apply Quadratic Formula

Now, substitute \(a = 1\), \(b=-6\), and \(D=-200\) into the quadratic formula.
Since \(\sqrt{-200}=\sqrt{200}\times\sqrt{-1}=10\sqrt{2}i\) (where \(i=\sqrt{-1}\) is the imaginary unit), we have:
\(x=\frac{-(-6)\pm10\sqrt{2}i}{2\times1}=\frac{6\pm10\sqrt{2}i}{2}\)

Step4: Simplify the Expression

Simplify \(\frac{6\pm10\sqrt{2}i}{2}\) by dividing each term in the numerator by 2:
\(x = 3\pm5\sqrt{2}i\)

Answer:

\(3 + 5\sqrt{2}i\) and \(3 - 5\sqrt{2}i\)