Question

the graph of a parabola can show whether a quadratic equation has real solutions. put the following descriptions in order, from indicating two real solutions to indicating no real solutions.
(1 point)

• a parabola that touches the x-axis at a single point.
• a parabola that crosses the x-axis twice.
• a parabola that does not intersect the x-axis at all.

Explanation:

Step1: Recall parabola - x - axis intersection and solutions

A quadratic equation \(ax^{2}+bx + c = 0\) (\(a
eq0\)) has solutions related to the intersection of the parabola \(y=ax^{2}+bx + c\) with the \(x\) - axis. If the parabola crosses the \(x\) - axis twice, the quadratic equation has two distinct real solutions. If the parabola touches the \(x\) - axis at a single point, the quadratic equation has one real solution (a repeated root). If the parabola does not intersect the \(x\) - axis at all, the quadratic equation has no real solutions.

Step2: Order the descriptions

We need to order from indicating two real solutions to indicating no real solutions.

• "A parabola that crosses the x - axis twice" indicates two real solutions.
• "A parabola that touches the x - axis at a single point" indicates one real solution (between two real solutions and no real solutions in terms of the number of solutions).
• "A parabola that does not intersect the x - axis at all" indicates no real solutions.

So the order is: 1. A parabola that crosses the x - axis twice; 2. A parabola that touches the x - axis at a single point; 3. A parabola that does not intersect the x - axis at all.

Answer:

1. A parabola that crosses the x - axis twice
2. A parabola that touches the x - axis at a single point
3. A parabola that does not intersect the x - axis at all