Question

Solve: 9x^2+2x= -3 , using the quadratic formula

Answer 1

SOLUTION

Given the question in the question tab, the following are the solution steps to answer the question.

STEP 1: Write the given quadratic equation.

$9x^2+2x=-3$

STEP 2: Express the equation in the standard quadratic form

$\begin{gathered} \text{standard form}=ax^2+bx+c=0 \\ 9x^2+2x=-3 \\ \text{Add 3 to both sides} \\ 9x^2+2x+3=-3+3 \\ 9x^2+2x+3=0 \end{gathered}$

STEP 3: Write the quadratic formula

$x_1,x_2=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}$

STEP 4: Write the needed parameters to substitute into the formula

$\begin{gathered} 9x^2+2x+3=0 \\ a=9,b=2,c=3 \end{gathered}$

STEP 5: Substitute the values into the formula and solve for x

$\begin{gathered} x_{1,2}=\frac{-2\pm\sqrt{2^2-4\cdot\:9\cdot\:3}}{2\cdot\:9} \\ \text{simplify }\sqrt[]{2^2-4\cdot\: 9\cdot\: 3} \\ \sqrt[]{2^2-4\cdot\: 9\cdot\: 3}=\sqrt[]{4-108}=\sqrt[]{-104}=2\sqrt{26}i \\ x_{1,\: 2}=\frac{-2\pm\sqrt[]{2^2-4\cdot9\cdot3}}{2\cdot\: 9}=\frac{-2\pm\:2\sqrt{26}i}{2\cdot\:9} \\ \mathrm{Separate\: the\: solutions} \\ x_1=\frac{-2+2\sqrt{26}i}{2\cdot\:9},\: x_2=\frac{-2-2\sqrt{26}i}{2\cdot\:9} \\ \frac{-2+2\sqrt[]{26}i}{2\cdot\: 9}=-\frac{1}{9}+\frac{\sqrt[]{26}}{9}i_{} \\ \frac{-2-2\sqrt[]{26}i}{2\cdot\: 9}=-\frac{1}{9}-\frac{\sqrt[]{26}}{9}i \\ x=-\frac{1}{9}+\frac{\sqrt[]{26}}{9}i\text{ or }-\frac{1}{9}-\frac{\sqrt[]{26}}{9}i \end{gathered}$

Hence, the roots of the equations are:

$\begin{gathered} x_1=-\frac{1}{9}+\frac{\sqrt[]{26}}{9}i \\ x_2=-\frac{1}{9}-\frac{\sqrt[]{26}}{9}i \end{gathered}$