Question

enter the correct answer in the box.
what are the solutions of this quadratic equation?
x² + 164 = 16x
substitute the values of a and b to complete the solutions.
(there is a math input panel and the form of solutions: x = a + bi, x = a - bi)

Explanation:

Step1: Rewrite the equation in standard form

The quadratic equation is given as \(x^{2}+164 = 16x\). We rewrite it in the standard form \(ax^{2}+bx + c=0\) by subtracting \(16x\) from both sides:
\(x^{2}-16x + 164=0\)
Here, \(a = 1\), \(b=- 16\) and \(c = 164\).

Step2: Use the quadratic formula

The quadratic formula for a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)

Substitute \(a = 1\), \(b=-16\) and \(c = 164\) into the formula:
First, calculate the discriminant \(\Delta=b^{2}-4ac\)
\(\Delta=(-16)^{2}-4\times1\times164\)
\(=256 - 656\)
\(=- 400\)

Now, find \(x\):
\(x=\frac{-(-16)\pm\sqrt{-400}}{2\times1}=\frac{16\pm\sqrt{400}\times\sqrt{- 1}}{2}\)
Since \(\sqrt{-1}=i\) and \(\sqrt{400} = 20\), we have:
\(x=\frac{16\pm20i}{2}\)
Simplify the fraction:
\(x = 8\pm10i\)

Answer:

For \(x=a + bi\), \(a = 8\) and \(b = 10\); for \(x=a - bi\), \(a=8\) and \(b = 10\). So the solutions are \(x = 8+10i\) and \(x=8 - 10i\)