Question

question
what are the roots of the equation
$4x^2 = 17x - 20$ in simplest $a + bi$ form?
answer

Explanation:

Step1: Rewrite the equation in standard form

First, we rewrite the given equation \(4x^{2}=17x - 20\) into the standard quadratic form \(ax^{2}+bx + c = 0\).
Subtract \(17x\) and add \(20\) to both sides:
\(4x^{2}-17x + 20=0\)
Here, \(a = 4\), \(b=- 17\), \(c = 20\).

Step2: Use the quadratic formula

The quadratic formula for the roots of a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac\).
Substitute \(a = 4\), \(b=-17\), \(c = 20\) into the discriminant formula:
\(\Delta=(-17)^{2}-4\times4\times20\)
\(=289 - 320\)
\(=- 31\)

Step3: Find the roots

Now, substitute \(a = 4\), \(b=-17\), and \(\Delta=-31\) into the quadratic formula:
\(x=\frac{-(-17)\pm\sqrt{-31}}{2\times4}=\frac{17\pm i\sqrt{31}}{8}\)
We can split this into two roots:
\(x_{1}=\frac{17}{8}+\frac{\sqrt{31}}{8}i\) and \(x_{2}=\frac{17}{8}-\frac{\sqrt{31}}{8}i\)

Answer:

The roots of the equation are \(\frac{17}{8}+\frac{\sqrt{31}}{8}i\) and \(\frac{17}{8}-\frac{\sqrt{31}}{8}i\)