A squared term signals a nonlinear system: a curve and a line may meet more than once. Graph both equations in Desmos and click every intersection, since each meeting point satisfies both equations. Then take the -coordinate from each point and multiply. If a question asks only for the product, substituting to get a quadratic offers another route.
Hints
- Hint 1
An intersection is a point on both graphs, so its coordinates make both equations true. A line can cross a parabola twice. Graph the two equations and look for every place they meet.
- Hint 2
A point is written , with first. Read the first coordinate of each intersection. The product means multiply those two values, not add them or multiply the -coordinates.
Step-by-step
Approach 1: Graph the system and multiply
Step 1Find every solution point
Type and on separate lines, then click both intersections. Desmos shows and . An intersection lies on both graphs, so each point satisfies the system. The first coordinate of each point is a possible value of .
- Step 2
Multiply the requested values
Multiply the first coordinates, and , keeping the negative sign:
Using the second coordinates would give the product of the -values instead. The product of all possible -values is . Grid in -6.
Approach 2: Use the product of quadratic roots
Step 1Replace one variable
Use substitution: replace in with from the first equation. Both equations must hold at the same point, so:
- Step 2
Remove the parentheses
Remove the parentheses and put the squared term first:
- Step 3
Write a quadratic equal to zero
Subtract from both sides:
The roots are the -values that make this equation equal .
- Step 4
Read the product from the coefficients
Call the roots and . In , the constant term is . So the product of the roots is . Here and . Substitute those coefficients:
The product of all possible -values is . Grid in -6.