Question

Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. $$\frac{1}{2} x^{2}+3=8$$

Short answer

The solutions to the equation are \(x\approx 3.16\) and \(x \approx -3.16\)

Step-by-step solution

Rearranging the equation

The given equation is \(\frac{1}{2} x^{2}+3=8\). First, we'll subtract 3 from both sides to isolate the term with \(x^{2}\). This gives us \(\frac{1}{2} x^{2} = 8 - 3\), which simplifies to \(\frac{1}{2} x^{2} = 5\).

Solve for \(x^{2}\)

Next, to solve for \(x^{2}\), we'll need to multiply both sides of the equation by 2. This gives us \(x^{2} = 5 \times 2\), which simplifies to \(x^{2} = 10\).

Solve for \(x\)

Finally, to solve for \(x\), we'll need to take the square root of both sides. Note that when we take the square root of \(x^{2}\), we get two solutions: \(+x\) and \(-x\). Therefore, \(x = \sqrt{10}\) and \(x = -\sqrt{10}\). Use a calculator to find the square roots, rounding to the nearest hundredths, if necessary.