Question

solve the equation using the quadratic formula.
x² = 10x - 61

Explanation:

Step1: Rewrite in standard form

First, we rewrite the equation \(x^2 = 10x - 61\) in the standard quadratic form \(ax^2+bx + c = 0\). Subtract \(10x\) and add \(61\) to both sides:
\(x^2-10x + 61=0\)
Here, \(a = 1\), \(b=- 10\), and \(c = 61\).

Step2: Apply quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b=-10\), and \(c = 61\) into the formula:
First, calculate the discriminant \(\Delta=b^2-4ac=(-10)^2-4\times1\times61=100 - 244=- 144\)
Then, \(x=\frac{-(-10)\pm\sqrt{-144}}{2\times1}=\frac{10\pm12i}{2}\) (since \(\sqrt{-144}=\sqrt{144}\times\sqrt{-1}=12i\))
Simplify the fraction: \(x = 5\pm6i\)

Answer:

\(x = 5 + 6i\) or \(x = 5-6i\)