Question

Solve the equation algebraically. Check the solutions graphically. $$\frac{2}{3} x^{2}=6$$

Short answer

The solutions to the equation \(\frac{2}{3}x^2 = 6\) are \(x = 3\) and \(x = -3\).

Step-by-step solution

Rearrange the Equation

Firstly, clear the fraction by multiplying both sides of equation by 3. This gives \(2x^2 = 18\).

Solve for x

Next, to isolate \(x\), firstly divide both sides of the equation by 2, giving \(x^2 = 9\). The solution for \(x\) is found by taking the square root of both sides. Remembering that the square root has two values, positive and negative, the two solutions for \(x\) are hence \(x = 3\) and \(x = -3\).

Check Solution Graphically

Graphically, the function \(y = \frac{2}{3}x^2\), is a parabola that opens upwards starting at the origin of the coordinate axes. Drawing vertical lines at \(x = 3\) and \(x = -3\) will intersect this parabola at \(y = 6\), therefore confirming the solutions found algebraically.