Question

complex numbers & higher order polynomials quick check
djamila is looking for conjugate pair solutions to the equation ( 3x^2 - 5x = -6 ). which of the following options should she choose? (1 point)
( circ \frac{5}{6} pm \frac{5}{6}i )
( circ \frac{5}{6} pm \frac{7}{6}i )
( circ \frac{5}{6} pm \frac{sqrt{47}}{6}i )
( circ -\frac{5}{6} pm \frac{sqrt{47}}{6}i )

Explanation:

Step1: Rewrite the equation

First, rewrite the quadratic equation \(3x^{2}-5x = - 6\) in standard form \(ax^{2}+bx + c=0\). Add 6 to both sides: \(3x^{2}-5x + 6=0\). Here, \(a = 3\), \(b=-5\), \(c = 6\).

Step2: Use the quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Calculate the discriminant \(\Delta=b^{2}-4ac\). Substitute \(a = 3\), \(b=-5\), \(c = 6\) into the discriminant formula: \(\Delta=(-5)^{2}-4\times3\times6=25 - 72=- 47\).

Step3: Substitute into quadratic formula

Now substitute \(a = 3\), \(b=-5\), \(\Delta=-47\) into the quadratic formula: \(x=\frac{-(-5)\pm\sqrt{-47}}{2\times3}=\frac{5\pm\sqrt{47}i}{6}=\frac{5}{6}\pm\frac{\sqrt{47}}{6}i\).

Answer:

\(\boldsymbol{\frac{5}{6}\pm\frac{\sqrt{47}}{6}i}\) (the third option)