A line and a parabola can meet twice, once, or not at all. Solving the system finds those points.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The method
Substitute the linear equation into the quadratic. One quadratic equation in one variable remains.
Then solve as usual
Rearrange to equal zero, then factor or use the formula. The discriminant now predicts how many intersections there are.
Answers come in pairs
Each x must be paired with its y. Substitute back into the linear equation, which is the easier one.
One solution means tangency
A single repeated solution means the line touches the parabola at exactly one point without crossing it.
Check both equations
A solution must satisfy both. Checking only the quadratic can let a slip through.
Substitute and solve
Where a line meets a parabola, substitute the linear expression into the quadratic. The result is a single quadratic equation, which is solved by the methods already available.
Two, one, or no intersections
The line can cut the parabola twice, touch it once as a tangent, or miss entirely. The discriminant of the resulting quadratic decides which, without solving it.
Find both coordinates
Solving gives the x-values; substituting back into the linear equation gives the matching y-values. An answer listing only x-values has not given the intersection points.
Tangency is the discriminant equalling zero
Setting the discriminant to zero and solving for an unknown coefficient is how you find the value making a line tangent to a curve. It is a standard and genuinely useful application.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Marcus Intersect a Line and a Curve
Marcus solves y = x² and y = x + 6.
- Step 1
Both equal y, so he sets them equal: x² = x + 6.
Step 4: Your Turn
Practice makes it stick.
The Intersection
Problem 1 of 2
y = x² and y = x + 6. What is the larger x value?
The Matching y
Problem 2 of 2
Same system. What is y when x = 3?
Line Meets Curve
1 of 8
y = x² and y = 4. What is the positive x value?
2 of 8
y = x² and y = 9. Positive x value?
3 of 8
y = x² and y = 0. How many intersection points?
4 of 8
y = x² and y = −4. How many intersection points?
5 of 8
y = x² and y = 2x. What is the non-zero x value?
6 of 8
y = x² and y = x + 6. What is the smaller x value?
7 of 8
Put the solving process in order.
- 1Rearrange so one side is zero.
- 2Solve the resulting quadratic for x.
- 3Substitute each x back to find its y.
- 4Substitute the linear equation into the quadratic.
8 of 8
y = x² and y = 25. Positive x value?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = x² and y = 16. What is the positive x value?
Question 2 of 2
What does exactly one solution mean geometrically?
What You Learned
- Substitute the linear equation into the quadratic and solve.
- A line and a parabola meet twice, once, or not at all.
- Pair each x with its y before reporting the answer.