Question

Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. $$\frac{2}{3} n^{2}-6=2$$

Short answer

The solutions are \(n\approx3.46\) and \(n\approx-3.46\), rounded to the nearest hundredth.

Step-by-step solution

Write Down the Equation

Firstly, the equation given in the exercise is \(\frac{2}{3} n^{2}-6=2\)

Simplifying the Equation

Now, begin by simplifying the equation. Add 6 to both sides to get rid of the negative constant on the left side of the equation, which results in: \(\frac{2}{3} n^{2} = 8\). Next, to isolate the term containing \(n^2\), you can multiply everything by \(\frac{3}{2}\), giving you the updated equation \(n^2 = 12\).

Calculating the Value of n

Next, to find the value of \(n\), take the square root of both sides. Remember, a square root has both a positive and a negative solution, so our results are \(n=\sqrt{12}\) and \(n=-\sqrt{12}\).

Rounding the Solution

After solving for \(n\), the results must be rounded to the nearest hundredth using a calculator. This gives \(n\approx3.46\) and \(n\approx-3.46\).