Line–quadratic intersections
Objective. Solve and interpret line and parabola intersections.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Line–quadratic intersections are found by substituting the linear equation into the quadratic equation, setting the equations equal, and solving the resulting quadratic for the x-coordinates where the line meets the parabola. Each valid solution for x is then substituted back into the linear equation to find the corresponding y-coordinate, giving the coordinates of each intersection point. The discriminant of the resulting quadratic reveals whether the line crosses the curve twice, touches it once as a tangent, or misses it entirely. For example, to intersect y=x+1 with y=x²−2x−3, solve x²−2x−3=x+1, giving x²−3x−4=0.
Worked example
Find the coordinates where the line y=2x−1 intersects the curve y=x²−4.
- 1Set the equations equal: x²−4 = 2x−1
- 2Rearrange: x²−2x−3=0, which factorises to (x−3)(x+1)=0
- 3Solve for x: x=3 or x=−1
- 4Substitute into y=2x−1: when x=3, y=5; when x=−1, y=−3
Answer: (3, 5) and (−1, −3)
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Start practisingCommon mistake. A common error is solving for x and forgetting to substitute back to find y, leaving the answer as just x-values; coordinates need both x and y, so always substitute each x back into the simpler linear equation.
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