GCSEGeniusMaths

GCSE Maths revision · Higher: number and algebra

Quadratic simultaneous equations

Higher only

The idea

With one linear and one quadratic equation, substitute the linear one into the quadratic. You get a quadratic in one letter, which usually gives two pairs of answers: the two crossing points.

  1. Rearrange the linear equation: y=…y = \dots or x=…x = \dots.
  2. Substitute into the quadratic and solve.
  3. Put each value back into the linear equation to find its partner.

Watch out Give both pairs, matched up correctly.

See it worked

Worked exampleCalculator allowed3 marks

Solve y = 2x + 1 and y = x2x^{2} −- 2 simultaneously. Give both ordered pairs (x, y).

  1.  x2−2=2x+1x^{2}-2=2x+1
  2.  (x−3)(x+1)=0(x-3)(x+1)=0
  3. Equate the expressions for y: 2x + 1 = x2x^{2} −- 2.
  4. Rearrange and factorise: x2x^{2} −- 2x −- 3 = (x −- 3)(x + 1) = 0.
  5. The x-values are 3 and −1-1; substituting into y = 2x + 1 gives y = 7 and −1-1 respectively.

Answer {(−1, −1), (3, 7)}

Your turn

Question 1Non-calculator6 marks

The curve y=x2y=x^2 meets the line y=6x−8y=6x-8.

(a) Find both intersection points exactly.

(b) Find the gradient of the chord joining the intersections.

Write your answer, then show the working
  1. (a) Equate the two expressions for y. x2=6x−8x^{2}=6x-8
  2. Factorise to obtain both x values. (x−2)(x−4)=0(x-2)(x-4)=0
  3. Use the original curve to recover the matching y values. y=22y=2^{2}
  4. Therefore (2,4)(2,4) and (4,16)(4,16).
  5. (b) Use rise over run from the two points. 16−44−2\frac{16-4}{4-2}
  6. Therefore 66.

Answer (a) (2,4)(2,4) and (4,16)(4,16); (b) 66

Question 2Non-calculator4 marks

The line y=2x+ky = 2x + k touches the curve y=x2+3y = x^2 + 3 at exactly one point.

(a) Find the value of kk.

(b) Write down the coordinates of the point where the line touches the curve.

Question 3Non-calculator3 marks

The line y = kx −- 2 meets the curve y = x2x^{2} −- 4x + 7 at exactly one point. Find both possible values of k.

In past papers

24 official questions in 24 papers since 2017. The latest:

All GCSE Maths past papers

GCSE Maths Pro

GCSE Maths Pro

£119 once

At least 12 months of access

By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.
Start free

No VAT is charged.