A line and a parabola, the graph of an equation with , may meet twice. Setting their expressions for equal gives a quadratic whose roots are the possible -values. Write it as , then use for the sum of the roots. Keep the minus sign, and don’t stop after finding only one value.
Hints
- Hint 1
A system solution is a point that satisfies both equations. At that point, both right sides give the same -value, so they must equal each other. What equation in does that give you?
- Hint 2
A quadratic in standard form looks like . Move the terms until one side is , keeping the sign of each term as you go.
- Hint 3
The root-sum rule says that the solutions to add to . Use the middle coefficient with its sign; you don’t need to find the two solutions separately.
Step-by-step
Approach 1: Use the sum of the roots
Step 1Set the two expressions for y equal
No Desmos needed. The root-sum rule gives the requested sum without finding both roots.
At a shared point, both equations give the same , so their right sides must be equal:
- Step 2
Bring the x-terms together
Subtract from both sides:
Keep the minus sign when moving the line’s term.
- Step 3
Put the quadratic in standard form
To get zero on the right, add to both sides:
- Step 4
Read the sum from the coefficients
For , the middle coefficient is times the roots’ sum. So that sum is . Here and :
When a system asks for the sum of its -values, use the root sum of the equation formed by setting the two -expressions equal. The sum of all possible -values is . Choice C.
Approach 2: Check the intersections in Desmos
Step 1Find both shared points
Type each equation on its own line in Desmos and click both intersections. Desmos shows and . Each intersection satisfies both equations, and its first coordinate is the -coordinate.
- Step 2
Add the requested coordinates
The first coordinates are and , so . The second coordinates are -values, not the values the question asks you to add. The sum of all possible -values is . Choice C.