A student solves the system y = x² and y = 3x − 4 by substitution, obtaining x² − 3x + 4 = 0. She computes the discriminant as 9 − 16 = −7 and concludes there must be an error in her algebra. What is wrong with her reasoning?
AShe made an error — the substitution should have produced a factorable quadratic
BA negative discriminant is a valid result meaning the line and parabola do not intersect in the real plane; no algebraic error has occurred
CShe should have used elimination instead of substitution to avoid this problem
DA negative discriminant means the system has infinitely many solutions
A negative discriminant is not an error — it is the algebraic signal that two curves do not intersect in the real plane. Geometrically, the line misses the parabola entirely. Assuming every algebraic setup must produce real solutions is the misconception; systems with no real solution are perfectly valid and important to recognize.
Question 2 Multiple Choice
After substituting into a nonlinear system and solving, a student finds x = 3 and x = −2. She reports the solutions as 'x = 3 and x = −2.' What critical step has she omitted?
AShe should have verified that her quadratic factors correctly
BShe should have drawn the graph to confirm the solutions exist
CShe must substitute each x-value back into an equation to find the corresponding y-values and report complete ordered pairs (3, y₁) and (−2, y₂)
DShe should check whether the discriminant is positive before accepting solutions
Solving for x gives the x-coordinates of the intersection points, but a solution to a system is a complete (x, y) pair. To find y, substitute each x-value back into either original equation. Reporting only x-values gives half-answers — you cannot plot or verify a solution that is missing a coordinate.
Question 3 True / False
A line and a parabola usually intersect in exactly two points because together they produce a quadratic equation, which usually has two solutions.
TTrue
FFalse
Answer: False
A quadratic equation has 0, 1, or 2 real solutions depending on the sign of the discriminant. A negative discriminant means the line and parabola don't intersect; a zero discriminant means exactly one intersection (tangency); a positive discriminant gives two intersections. 'Quadratic equation' does not guarantee two real solutions.
Question 4 True / False
Sketching the graphs of a nonlinear system before solving algebraically is useful because it lets you predict how many solutions to expect and provides a visual check on algebraic results.
TTrue
FFalse
Answer: True
Geometry precedes algebra here: the number of intersections visible in a sketch (0, 1, or 2 for a line-parabola system) should match the number of real solutions your algebra produces. If the graph suggests two intersections but algebra yields a negative discriminant, you know to look for errors. This geometric preview is part of the complete solution toolkit.
Question 5 Short Answer
Why does the discriminant of the quadratic produced by substitution tell you how many solutions a line-parabola system has?
Think about your answer, then reveal below.
Model answer: Substituting the linear equation into the quadratic yields a single-variable quadratic equation. Its real solutions correspond exactly to x-coordinates where the two curves intersect. The discriminant b² − 4ac determines the number of real roots: positive gives two intersections, zero gives one (tangency), negative gives no real intersection. The algebraic count of solutions mirrors the geometric count of intersection points.
This connection between algebra and geometry is the key insight of nonlinear systems: the discriminant is not just an algebraic calculation — it encodes whether the two curves actually meet in the real plane. Understanding this makes the case of 'no solution' just as meaningful as cases with one or two solutions.