Algebra 9
3 marks · NormalExplain, using algebraic methods, why the system below can have zero, one or two solutions:
where c is a number.
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.
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Must show: Reference to discriminant/quadratic equation, Clear explanation for 0, 1, 2 solutions
Also accept: If discriminant > 0: 2 solutions; if =0: 1 solution; if <0: 0 solutions.
Common mistakes: No working or incomplete justification
Show mark scheme and worked solution
Marking points
B1 · 1 mark · Forms quadratic equation by equating y expressions.
B2 · 1 mark · Explains that the number of solutions depends on the discriminant (b²-4ac) of the quadratic equation in x.
B3 · 1 mark · States that two intersections (2 solutions) occur if the line cuts the curve twice, one solution if the line is tangent, and none if it does not meet.
Worked solution
Set the two equations equal: x^2 + 3x = 2x + c
Rearrange: x^2 + x - c = 0
The number of solutions depends on the discriminant: D = 1^2 - 4(1)(-c) = 1 + 4c
If D > 0, two real solutions (intersection at two points); if D = 0, one solution (tangent); if D < 0, no solution (no intersection).