Question

Solve the equation algebraically. Check the solutions graphically. $$ \frac{1}{3} x^{2}=3 $$

Short answer

The algebraic solution to the equation \( \frac{1}{3} x^{2}=3 \) are \( x = \pm 3 \), these are also confirmed when checking graphically.

Step-by-step solution

Write the Equation in Standard Form

First transform the given equation into the standard form for quadratic equations. In this case, it means multiplying both sides of \( \frac{1}{3} x^{2}=3 \) by 3. Which outputs: \( x^{2}=9 \)

Take the Square Root of Both Sides

In order to find the roots or zeroes of the equation, the square root of both sides of the equation should be taken. So, \( sqrt{x^{2}} = \pm sqrt{9} \). This leads to: \( x = \pm 3 \)

Graph the Equation

Now, confirm the solutions graphically by plotting the original equation which will take a form of a parabola. Then check where it crosses the x-axis. As accurately as possible, based on the graph the intersection will be at -3 and 3, which confirms the solutions obtained algebraically.