Question

Hi I need help with this question please and thank you

Answer 1

Statement Problem: Given the equation,

$3x^2+5x+3=0$

What is the solution(s)?

Solution:

Step 1: Divide through by 3;

$\begin{gathered} \frac{3x^2}{3}+\frac{5x}{3}+\frac{3}{3}=\frac{0}{3} \\ x^2+\frac{5}{3}x+1=0 \end{gathered}$

Step 2: Subtract 1 from both sides of the equation,

$\begin{gathered} x^2+\frac{5}{3}x+1-1=0-1 \\ x^2+\frac{5}{3}x=-1 \end{gathered}$

Step 3: Add the square of half of the coeffient of x to both sides of the equation;

$\begin{gathered} x^2+\frac{5}{3}x+(\frac{1}{2}(\frac{5}{3}))^2=-1+(\frac{1}{2}(\frac{5}{3}))^2 \\ x^2+\frac{5}{3}x+(\frac{5}{6})^2=-1+(\frac{5}{6})^2 \\ x^2+\frac{5}{3}x+(\frac{5}{6})^2=-1+\frac{25}{36} \end{gathered}$

Step 4: Factorize the left side and simplify the right side,

$\begin{gathered} (x+\frac{5}{6})^2=\frac{-36+25}{36} \\ (x+\frac{5}{6})^2=-\frac{11}{36} \end{gathered}$

Step 5: Take square root of both sides,

$\begin{gathered} \sqrt[]{(x+\frac{5}{6})^2}=\sqrt[]{-\frac{11}{36}} \\ \sqrt[]{(x+\frac{5}{6})^2}=\frac{\sqrt[]{-11}}{\sqrt[]{36}} \\ x+\frac{5}{6}=\pm\frac{i\sqrt[]{11}}{6} \end{gathered}$

Step 6: Subtract 5/6 from both sides of the equation,

$\begin{gathered} x+\frac{5}{6}-\frac{5}{6}=-\frac{5}{6}\pm\frac{i\sqrt[]{11}}{6} \\ x=\frac{-5\pm i\sqrt[]{11}}{6} \\ x_1=\frac{-5+i\sqrt[]{11}}{6},x_2=\frac{-5-i\sqrt[]{11}}{6} \end{gathered}$