Cogito
Integrated Math 2 · Chapter 2 · Lesson 3
Systems With a Quadratic
Where a line meets a curve.
12 problems · about 21 minutes · A-REI.C.7
What this lesson teaches
The student solves systems containing a linear and a quadratic equation.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy = x² and y = 16. What is the positive x value?
Answer 4
Why 4.
What does exactly one solution mean geometrically?
Answer The line touches the parabola without crossing it.
Why The line is tangent to the curve.
y = x² and y = 4. What is the positive x value?
Answer 2
Why x² = 4.
y = x² and y = 9. Positive x value?
Answer 3
Why x² = 9.
y = x² and y = 0. How many intersection points?
Answer 1
Why The vertex touches the axis.
Build It Up
The same ideas with more to keep track of.
3 problemsy = x² and y = −4. How many intersection points?
Answer 0
Why A square is never negative.
y = x² and y = 2x. What is the non-zero x value?
Answer 2
Why x² − 2x = 0.
y = x² and y = x + 6. What is the smaller x value?
Answer -2
Why (x − 3)(x + 2).
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the solving process in order.
Answer 1. Substitute the linear equation into the quadratic. 2. Rearrange so one side is zero. 3. Solve the resulting quadratic for x. 4. Substitute each x back to find its y.
Why The y values come last.
y = x² and y = 25. Positive x value?
Answer 5
Why x² = 25.
The Intersection: y = x² and y = x + 6. What is the larger x value?
Answer 3
Why 3.
The Matching y: Same system. What is y when x = 3?
Answer 9
Why 9.