Question

1. (0 points) graph the system of inequalities. then find the coordinates of the points of intersection of the graphs of the related equations.

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$$\begin{cases} y \\ge x^2 - 3 \\\\ y \\le 2x \\end{cases}$$

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Explanation:

Step 1: Set equations equal

Set the related equations equal to find intersection points.

$$x^2 - 3 = 2x$$

Step 2: Rearrange the equation

Rearrange the terms into standard quadratic form.

$$x^2 - 2x - 3 = 0$$

Step 3: Factor the quadratic

Factor the quadratic equation to find the roots.

$$(x - 3)(x + 1) = 0$$

Step 4: Solve for x

Find the x-coordinates of the intersection points.

$$x = 3 \quad \text{or} \quad x = -1$$

Step 5: Find corresponding y-coordinates

Substitute the x-values back into the linear equation.

$$y = 2(3) = 6 \quad \text{and} \quad y = 2(-1) = -2$$

Answer:

The coordinates of the points of intersection are \((-1, -2)\) and \((3, 6)\).