A line meeting a parabola, step by step
A line crosses a parabola twice, and finding where is substitution rather than elimination.
The question
Find the x-values where y = x² + 2x − 23 meets y = x − 3.
Find the x-values where y = x² + 2x − 23 meets y = x − 3
x² + 2x − 23 = x − 3
x² + x − 20 = 0
(x + 5)(x − 4) = 0
x = −5x = 4
- Both equations say what y is, so where the graphs meet the two right-hand sides are equal: x² + 2x − 23 = x − 3.
- Bring everything to one side: x² + x − 20 = 0
- Factor it. Two numbers multiplying to −20 and adding to 1 give (x + 5)(x − 4) = 0
- So the graphs meet where x = −5 and x = 4.
How it works.
- 01
Both equations say what y is, so where the graphs meet the two right-hand sides are equal: x² + 2x − 23 = x − 3.
- 02
Bring everything to one side: x² + x − 20 = 0
- 03
Factor it. Two numbers multiplying to −20 and adding to 1 give (x + 5)(x − 4) = 0
- 04
So the graphs meet where x = −5 and x = 4.