Question

Solve the equation by graphing the related function.one halfx Superscript 2minus2equals0Question content area bottomPart 1Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.A.xequalsenter your response here ​(Simplify your answer. Use a comma to separate answers as​ needed.)B.The solution is not a real number.

Answer 1

To solve the equation \(\frac{1}{2}x^2 - 2 = 0\) by graphing the related function, we can first rewrite the equation as a function:

\[ f(x) = \frac{1}{2}x^2 - 2 \]

Next, we need to find the x-values where this function is equal to zero, since that corresponds to the solutions of the equation.

1. **Graph the function**:

The function \(f(x)\) is a quadratic function that opens upwards (since the coefficient of \(x^2\) is positive). It will have its vertex at the minimum point, which can be found using the vertex formula \(x = -\frac{b}{2a}\), where \(f(x) = ax^2 + bx + c\). In our case, \(a = \frac{1}{2}\), \(b = 0\), and \(c = -2\).

\[
x = -\frac{0}{2 \cdot \frac{1}{2}} = 0
\]

Plugging \(x = 0\) into the function:

\[
f(0) = \frac{1}{2}(0)^2 - 2 = -2
\]

This means the vertex is at the point \((0, -2)\).

2. **Find the x-intercepts**:

To find the x-intercepts, we set \(f(x)\) to zero:

\[
\frac{1}{2}x^2 - 2 = 0
\]

Multiplying both sides by 2 to eliminate the fraction:

\[
x^2 - 4 = 0
\]

Factoring the equation:

\[
(x - 2)(x + 2) = 0
\]

This gives us two solutions:

\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

Thus, the solutions to the equation \(\frac{1}{2}x^2 - 2 = 0\) are:

\[
x = 2, -2
\]

### Final Choice:
A. \(x = -2, 2\) (Simplified answer).