Algebra 9
4 marks · HardA student claims: "A pair of equations, one linear and one quadratic, must always have two solutions for ." Is the student correct? Explain using an example for each possibility, referring to the graphs of the equations.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
Must show: Correctly state the claim is false, A suitable pair with two solutions (e.g. intersects twice), A pair with one solution (e.g. tangent), A pair with no real solutions (e.g. line misses parabola)
Common mistakes: Only discussing one case; Not explaining graphically
Show mark scheme and worked solution
Marking points
B1 · 1 mark · States the claim is false with reasoning.
M1 · 1 mark · Provides example with two distinct solutions (line cuts parabola twice).
M2 · 1 mark · Provides example with one solution (line tangent to parabola).
M3 · 1 mark · Provides example with no solution (line does not intersect parabola).
Worked solution
No, the claim is not always true. There are three possible cases when you solve a linear and a quadratic equation simultaneously:
1. Two solutions: The line and the curve cross each other at two distinct points.
Example: , ⇒ ⇒ ; this quadratic has two real solutions.
2. One solution: The line just touches the parabola (is tangent to it).
Example: , ⇒ ⇒ ⇒ ; actually, let's adjust to get one repeated solution.
For instance, line , curve : ⇒ (Oops, this still gives two). Try and : ⇒ (still two). Let's use , : intersects at just one point at (0, 0), but this is not a tangent. The line , : — so only one solution, line is tangent.
3. No solution: The line and the parabola do not meet.
Example: , ⇒ ⇒ — this has no real roots since its discriminant is negative.
Graphically: - Two solutions = two intersection points. - One solution = the line is tangent to the parabola. - No solution = the line does not meet the parabola at all. Therefore, it's not true that every pair has two solutions.