Simultaneous Equations: Linear and Quadratic
Simultaneous equations with one linear and one quadratic equation are solved by substituting the linear equation into the quadratic one, producing a single equation in one variable, solved by factorising or the quadratic formula. Because this equation is quadratic, there are usually two solutions, giving two pairs of x and y values (or none, if the line misses the curve).
Before you start
Make sure you're comfortable with these topics first:
Method
- Rearrange the linear equation to make one variable the subject, for example y = ... or x = ...
- Substitute this expression into the quadratic equation in place of that variable, to eliminate it.
- Expand any brackets and simplify to form a quadratic equation equal to zero.
- Solve the quadratic equation by factorising, completing the square, or using the quadratic formula.
- Substitute each x-value back into the linear equation (not the quadratic one) to find the matching y-value.
- State both solutions as coordinate pairs, and check them in both original equations if time allows.
Worked example
Solve the simultaneous equations y = x + 2 and y = x^2 - 4x + 6.
- Substitute y = x + 2 into the quadratic equation: x + 2 = x^2 - 4x + 6.
- Rearrange into a quadratic equal to zero: x^2 - 5x + 4 = 0.
- Factorise: (x - 1)(x - 4) = 0, giving x = 1 or x = 4.
- Substitute each x-value into y = x + 2 to find y = 3 and y = 6.
- Final answer: (1, 3) and (4, 6).
Practice questions
Try each question, then tap to reveal the answer.
Q1Solve the simultaneous equations y = x + 1 and y = x^2 - 1.Show answer
Answer: (2, 3) and (-1, 0)
Q2Solve the simultaneous equations y = 3x - 2 and y = x^2 - 2x - 2.Show answer
Answer: (0, -2) and (5, 13)
Q3Solve the simultaneous equations x + y = 6 and y = x^2 - 2x.Show answer
Answer: (3, 3) and (-2, 8)
Q4Solve the simultaneous equations xy = 12 and x + y = 7.Show answer
Answer: (3, 4) and (4, 3)
Q5A curve has equation x^2 + y^2 = 25. A line has equation y = x + 1. Find the coordinates of the two points where the line intersects the curve.Show answer
Answer: (-4, -3) and (3, 4)
Q6Show that the line y = 2x + 5 does not intersect the curve y = x^2 + 3x + 9.Show answer
Answer: Rearranges to x^2 + x + 4 = 0; discriminant = 1 - 16 = -15 < 0, so no real solutions
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Solve the simultaneous equations y = x - 1 and y = x^2 - 5x + 7.
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A curve has equation x^2 + y^2 = 34. A line has equation y = x - 2. Find the coordinates of the two points where the line intersects the curve.
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The line y = 4x + c is a tangent to the curve y = x^2 + 2x + 5. Find the value of c.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the simultaneous equations: linear and quadratic worksheet pack - 11 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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