No Real Number Solutions

Overcoming a Third Obstacle to Factoring—What If There Are No Real Number Solutions?

Student Outcomes

• Students understand the possibility that an equation—or a system of equations—has no real solutions.
• Students identify these situations and make the appropriate geometric connections.

New York State Common Core Math Algebra II, Module 1, Lesson 36

Worksheets for Algebra II, Module 1, Lesson 36

Solutions for Algebra II, Module 1, Lesson 36

Classwork

Opening Exercise
Find all solutions to each of the systems of equations below using any method.
2𝑥 − 4𝑦 = −1
3𝑥 − 6𝑦 = 4

𝑦 = 𝑥² − 2
𝑦 = 2𝑥 − 5

𝑥² + 𝑦² = 1
𝑥² + 𝑦² = 4

Exercises 1–4

1. Are there any real number solutions to the system 𝑦 = 4 and 𝑥² +𝑦² = 2? Support your findings both analytically and graphically.
2. Does the line 𝑦 = 𝑥 intersect the parabola 𝑦 = −𝑥²? If so, how many times and where? Draw graphs on the same set of axes.
3. Does the line 𝑦 = −𝑥 intersect the circle 𝑥² + 𝑦² = 1? If so, how many times and where? Draw graphs on the same set of axes.
4. Does the line 𝑦 = 5 intersect the parabola 𝑦 = 4 − 𝑥²? Why or why not? Draw the graphs on the same set of axes.

Lesson Summary

An equation or a system of equations may have one or more solutions in the real numbers, or it may have no real number solution.

Two graphs that do not intersect in the coordinate plane correspond to a system of two equations without a real solution. If a system of two equations does not have a real solution, the graphs of the two equations do not intersect in the coordinate plane.

A quadratic equation in the form 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0, that has no real solution indicates that the graph of 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐 does not intersect the 𝑥-axis.

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