Question

complex numbers online practice
complete this assessment to review what youve learned. it will not count toward your grade.
solve the quadratic equation $12x^2 - 12x + 4 = -3x$. which of the following expresses its solutions in the form $a \pm bi$? (1 point)
\\(\frac{3}{8} \pm \frac{\sqrt{273}}{24}i\\)
\\(-\frac{3}{8} \pm \frac{\sqrt{111}}{24}i\\)
\\(\frac{3}{8} + \frac{\sqrt{111}}{24}i\\)
\\(\frac{3}{8} \pm \frac{\sqrt{111}}{24}i\\)

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the quadratic equation \(12x^{2}-12x + 4=-3x\) in standard form \(ax^{2}+bx + c = 0\).
Add \(3x\) to both sides: \(12x^{2}-12x + 3x+4 = 0\), which simplifies to \(12x^{2}-9x + 4=0\). Here, \(a = 12\), \(b=-9\), \(c = 4\).

Step2: Use the quadratic formula

The quadratic formula for a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac\). Substitute \(a = 12\), \(b=-9\), \(c = 4\) into the discriminant formula:
\(\Delta=(-9)^{2}-4\times12\times4=81 - 192=- 111\)
Then, substitute \(a = 12\), \(b=-9\), and \(\Delta=-111\) into the quadratic formula:
\(x=\frac{-(-9)\pm\sqrt{-111}}{2\times12}=\frac{9\pm i\sqrt{111}}{24}\) (since \(\sqrt{-111}=i\sqrt{111}\))
Simplify \(\frac{9}{24}=\frac{3}{8}\), so \(x = \frac{3}{8}\pm\frac{\sqrt{111}}{24}i\)

Answer:

\(\frac{3}{8}\pm\frac{\sqrt{111}}{24}i\) (the last option: \(\boldsymbol{\frac{3}{8}\pm\frac{\sqrt{111}}{24}i}\))