Solve a linear and a quadratic equation together by substituting, then find every matching pair of values. It is Higher only, aimed at grade 8. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A line and a curve: substitute the linear expression into the quadratic (or set two y= expressions equal), collect =0, solve the quadratic, then find each matching y from the line. Expect two pairs, one pair (tangent) or none. Expand (x+1)2 fully — three terms, not two.
Present answers as pairs or coordinates. Check both pairs in the equation you did not substitute into. A circle x2+y2=r2 with a line is the same method; the algebra is just a bit thicker.
Substitute the linear into the quadratic (or equate two y= expressions). You will get a quadratic in one letter — factorise or use the formula. Each x gives a y. Two solutions means two pairs; write them as coordinates. A circle x2+y2=r2 with y=mx is x2(1+m2)=r2.
Objectives
Substitute the line into the curve.
Solve the quadratic you get.
Return coordinates, not bare x-values.
Key points
Rearrange a linear equation to make one letter the subject.
Substitute an expression into a quadratic and expand carefully, including brackets squared.
Collect everything on one side and factorise (or use the formula) to solve the quadratic.
Find the matching value of the other letter for each solution and write the answers in pairs.
Worked examples
Example 1
Where do y=x^2 and y=x+2 meet? Give both points.
2 marks
x^2=x+2
(x-2)(x+1)=0
(-1, 1) and (2, 4)
Answer: (-1, 1) and (2, 4)
Common slips
Slip: x^2=2 The line is x+2, not the number 2 alone.
Slip: (x-2)(x-1)=0 -2 and 1 are the pair that multiply to -2 and add to -1.
Slip: x=-1 and x=2 Find y from the line as well.
Example 2
Find the points where y=x^2 meets y=2x+3.
2 marks
x^2-2x-3=0
(x-3)(x+1)=0
(-1, 1) and (3, 9)
Answer: (-1, 1) and (3, 9)
Common slips
Slip: x^2+2x+3=0 Subtract 2x+3 from x^2.
Slip: (x-3)(x-1)=0 3 and -1 multiply to -3.
Slip: (3, 9) only x=-1 is a real meeting point.
Example 3
Solve the pair y=x^2+1 and y=2x+1.
3 marks
x^2+1=2x+1
x(x-2)=0
(0, 1) and (2, 5)
Answer: (0, 1) and (2, 5)
Common slips
Slip: x^2=2x+1 The curve is x^2+1, so the +1 stays until you subtract.
Slip: the only root is 2 x=0 also makes the product zero.
Slip: They touch at one point The x-values differ, so there are two points.
Questions and answers
GCSE Maths quadratic simultaneous equations questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Find the coordinates where y=x^2 meets y=3x-2.
Show answer
Answer: (1, 1) and (2, 4)
Set x^2=3x-2.
Factor (x-1)(x-2)=0.
M1 — Set x^2=3x-2.
A1 — Answer (1, 1) and (2, 4)
Question 2 Easy2 marks
List the intersections of y=x^2 and y=-x+6.
Show answer
Answer: (-3, 9) and (2, 4)
x^2+x-6=0.
(x+3)(x-2)=0.
M1 — x^2+x-6=0.
A1 — Answer (-3, 9) and (2, 4)
Question 3 Medium2 marks
y=x^2 cuts y=5x-6. Write the two points.
Show answer
Answer: (2, 4) and (3, 9)
x^2-5x+6=0.
(x-2)(x-3)=0.
M1 — x^2-5x+6=0.
A1 — Answer (2, 4) and (3, 9)
Question 4 Medium2 marks
Find where the line y=-2x+8 hits y=x^2.
Show answer
Answer: (-4, 16) and (2, 4)
x^2+2x-8=0.
(x+4)(x-2)=0.
M1 — x^2+2x-8=0.
A1 — Answer (-4, 16) and (2, 4)
Question 5 Hard2 marks
How many intersections do y=x^2 and y=-x-1 have?
Show answer
Answer: none
x^2+x+1=0.
The discriminant is 1-4, which is negative.
M1 — x^2+x+1=0.
A1 — Answer none
Question 6 Hard2 marks
The points (1, 1) and (3, 9) lie on y=x^2. Write the straight line through them.
Show answer
Answer: y=4x-3
Gradient is (9-1)/(3-1).
The line is y-1=4(x-1).
M1 — Gradient is (9-1)/(3-1).
A1 — Answer y=4x-3
Question 7 Exam-style5 marks
The line y=3x meets y=x^2+2. Find the points, and say whether the line is a tangent.
Show answer
Answer: (1, 3) and (2, 6)
x^2-3x+2=0.
(x-1)(x-2)=0.
Points (1, 3) and (2, 6).
Two distinct points, so it is not a tangent.
M1 — Substitute
M1 — Factor
A1 — First point
A1 — Second point
A1 — Not a tangent
Common mistakes
Most lost marks on this topic are one slip.
Giving only the two x-values
Each meeting point is a coordinate pair, so find y from the line
Using only the first root
A quadratic often has a second root that gives a second point
Sign slips when moving the line across
Subtract the whole line expression from the curve, then set it equal to 0
Spot the slip
Jo says y=x^2 and y=x+1 meet only at the origin. What did she skip?
She set x^2=0 instead of x^2=x+1
Lines never meet curves
She should have added the equations
Show the slip
She set x^2=0 instead of x^2=x+1 Put the line in place of y.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the substitution line in full with brackets before expanding — it usually earns a method mark on its own.
When the x2 coefficient is negative, multiply the whole equation by −1 before factorising to avoid sign slips.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.