Question

solve the equation $2x^2 = 6x - 5$. (1 point)
$\frac{3}{2} \pm \frac{1}{2}i$
$\frac{3}{2} \pm \frac{\sqrt{19}}{2}$
$-\frac{3}{2} \pm \frac{\sqrt{19}}{2}$
$-\frac{3}{2} \pm \frac{1}{2}i$

Explanation:

Step1: Rewrite the equation in standard form

First, we rewrite the given equation \(2x^{2}=6x - 5\) into the standard quadratic form \(ax^{2}+bx + c = 0\). Subtract \(6x\) and add \(5\) to both sides:
\(2x^{2}-6x + 5=0\)
Here, \(a = 2\), \(b=-6\), and \(c = 5\).

Step2: Apply the quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute \(a = 2\), \(b=-6\), and \(c = 5\) into the formula:
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-6)^{2}-4\times2\times5=36 - 40=- 4\)
Then, \(x=\frac{-(-6)\pm\sqrt{-4}}{2\times2}=\frac{6\pm2i}{4}\) (since \(\sqrt{-4}=\sqrt{4}\times\sqrt{-1} = 2i\))
Simplify the fraction: \(\frac{6\pm2i}{4}=\frac{3\pm i}{2}=\frac{3}{2}\pm\frac{1}{2}i\)

Answer:

\(\frac{3}{2}\pm\frac{1}{2}i\) (corresponding to the first option: \(\boldsymbol{\frac{3}{2}\pm\frac{1}{2}i}\))