Quadratic simultaneous equations

Quadratic simultaneous equations are a pair of equations where one is linear and the other is quadratic, such as y = x + 6 and x² + y² = 50. To solve them, rearrange the linear equation to make x or y the subject, substitute it into the quadratic, solve the quadratic that results, then substitute each answer back to find its partner. There are usually two pairs of solutions, the two points where a line crosses a curve.

What are quadratic simultaneous equations?

Ordinary simultaneous equations are two straight lines, which cross at most once. When one equation is quadratic, its graph is a curve: a parabola such as y = x² − 4x + 5, or a circle such as x² + y² = 50. A straight line can cross that curve twice, touch it once or miss it completely, so there can be two, one or no pairs of solutions.

Elimination does not work here, because the x² or y² terms will not cancel with a linear equation. Substitution is the method.

How to solve quadratic simultaneous equations

1. Rearrange the linear equation to get y = ... or x = .... Pick whichever avoids fractions.
2. Substitute it into the quadratic equation. Put brackets round what you substitute.
3. Expand and collect terms to make a quadratic equal to 0.
4. Solve the quadratic by factorising or with the quadratic formula. You usually get two values.
5. Substitute each value into the linear equation to find its partner. Always use the linear one: it gives one answer per value, while the quadratic can give two.
6. Write the answers in pairs and check one pair in both original equations.

Example 1: a line and a circle

Solve y = x + 6 and x² + y² = 50.

The linear equation already gives y. Substitute it into the circle:

So x = 1 or x = −7. Now use y = x + 6: when x = 1, y = 7; when x = −7, y = −1.

The solutions are x = 1, y = 7 and x = −7, y = −1: the points A and B in the diagram. Check: 1² + 7² = 50 and (−7)² + (−1)² = 50.

Example 2: a line and a parabola

Solve y = x² − 4x + 5 and y = 2x − 3.

Both equations give y, so set them equal:

So x = 2 or x = 4. Using y = 2x − 3: y = 1 or y = 5. The solutions are (2, 1) and (4, 5).

Example 3: rearranging first

Solve x + y = 3 and x² + 2y² = 9.

Rearrange the linear equation: x = 3 − y. Substitute:

So y = 0 or y = 2. Then x = 3 − y gives x = 3 or x = 1. The solutions are x = 3, y = 0 and x = 1, y = 2. Notice the quadratic had no number term, so it factorised with a common factor of 3y. Do not divide both sides by y, or you lose the solution y = 0.

Ordinary simultaneous equations come first. Solve this linear pair, then press Mark to check both values in the course.

Two solutions, one or none

After substituting, the quadratic tells you how the line and curve meet. Its discriminant, b² − 4ac, decides:

For example, substituting y = x + 4 into x² + y² = 8 gives 2x² + 8x + 8 = 0, which is 2(x + 2)² = 0. Here b² − 4ac = 8² − 4 × 2 × 8, which is zero. So there is one repeated solution, x = −2 and y = 2: the line just touches the circle at (−2, 2), as a tangent. A-level questions use this idea to find the value of a constant that makes a line a tangent.

Common mistakes

• Squaring a bracket wrongly. (x + 6)² is x² + 12x + 36, not x² + 36.
• Substituting back into the quadratic. That can give extra, wrong answers. Use the linear equation.
• Not pairing the answers. x = 1 goes with y = 7, and x = −7 with y = −1. Writing x = 1, −7 and y = 7, −1 without pairing can lose a mark.
• Dividing by x or y. It loses the solution where that letter is 0. Factorise instead.
• Forgetting the brackets when substituting an expression with a minus sign, such as (3 − y)².

Quadratic simultaneous equations at GCSE

In AQA's GCSE Maths specification, topic A19 covers solving two simultaneous equations in two variables algebraically and finding approximate solutions using a graph, including the case where one is linear and one is quadratic. Linear and quadratic pairs are on Higher papers. They come back at A-level in coordinate geometry, for lines meeting circles and curves.

Practise in a short timed paper

GCSE Maths Higher and A-level Maths: Algebra and Functions each give you a short timed paper every day, with 8 questions worth 25 to 35 marks. The Simultaneous Equations Calculator checks your answers and draws the graphs.

Read next

Maths · AlgebraSimultaneous equations: how to solve them, with examplesHow to solve simultaneous equations by elimination, substitution and graphs, with worked GCSE examples, word problems and practice questions with answers.