Question

two solutions for all quadratic equations practice
complete this assessment to review what youve learned. it will not count toward your grade.

solve the quadratic equation $3x^2 + 3x + 12 = 10$. express its solutions in the form $a \pm bi$. (1 point)

check answer remaining attempts: 3

Explanation:

Step1: Simplify the equation

First, we need to rewrite the quadratic equation in standard form \(ax^{2}+bx + c = 0\). Given \(3x^{2}+3x + 12=10\), subtract 10 from both sides:
\(3x^{2}+3x + 12 - 10=0\)
\(3x^{2}+3x + 2 = 0\)

Step2: Identify coefficients

For the quadratic equation \(ax^{2}+bx + c = 0\), here \(a = 3\), \(b = 3\), \(c = 2\).

Step3: Use quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). First, calculate the discriminant \(\Delta=b^{2}-4ac\).
Substitute \(a = 3\), \(b = 3\), \(c = 2\) into the discriminant formula:
\(\Delta=(3)^{2}-4\times3\times2=9 - 24=- 15\)

Step4: Substitute into quadratic formula

Now substitute \(a = 3\), \(b = 3\), \(\Delta=-15\) into the quadratic formula:
\(x=\frac{-3\pm\sqrt{-15}}{2\times3}\)
Since \(\sqrt{-15}=\sqrt{15}\times\sqrt{-1}=i\sqrt{15}\) (where \(i\) is the imaginary unit, \(i^{2}=-1\)):
\(x=\frac{-3\pm i\sqrt{15}}{6}\)
Simplify the fraction:
\(x=-\frac{1}{2}\pm\frac{\sqrt{15}}{6}i\)

Answer:

\(-\frac{1}{2}\pm\frac{\sqrt{15}}{6}i\)