Question

solve the system of equations by graphing. first graph the equations, and then fill in the solutions.
$y = \frac{1}{4}x^2 - 2$
$y = \frac{1}{2}x$
to graph a quadratic equation, click to plot the vertex first.
to graph a linear equation, click to plot points on the graph.
the solutions are (\square,\square) and (\square,\square).

Explanation:

Step1: Set equations equal

To find the solutions, set \(\frac{1}{4}x^{2}-2=\frac{1}{2}x\).
Multiply both sides by 4 to eliminate fractions: \(x^{2}-8 = 2x\).
Rearrange into standard quadratic form: \(x^{2}-2x - 8=0\).

Step2: Factor the quadratic

Factor \(x^{2}-2x - 8\). We need two numbers that multiply to -8 and add to -2. These numbers are -4 and 2.
So, \(x^{2}-2x - 8=(x - 4)(x + 2)=0\).

Step3: Solve for x

Set each factor equal to zero:

• \(x - 4=0\) gives \(x = 4\).
• \(x + 2=0\) gives \(x=-2\).

Step4: Find corresponding y-values

For \(x = 4\), substitute into \(y=\frac{1}{2}x\): \(y=\frac{1}{2}(4)=2\).
For \(x=-2\), substitute into \(y=\frac{1}{2}x\): \(y=\frac{1}{2}(-2)=-1\).

Answer:

The solutions are \((4, 2)\) and \((-2, -1)\).