Studyledge

Quadratic Equations: O-Level / IGCSE Maths (0580)

Syllabus E2.4, E2.5 · Strand 2 Algebra and graphs

Questions
4
Total marks
20
Tier mix
0 Core · 4 Extended

A quadratic equation can always be written in the form

ax2+bx+c=0,a≠0.ax^{2} + bx + c = 0, \qquad a \neq 0.

On the Extended 0580 paper there are three standard ways to solve one, and a strong answer chooses the quickest valid method for the numbers in front of you:

  • Factorising: fastest when the quadratic factorises with whole numbers.
  • The quadratic formula   x=−b±b2−4ac2a  \;x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\; always works, and is the method to reach for when the equation does not factorise (the answer is then usually required to a stated number of decimal places).
  • Completing the square rewrites the equation as (x+p)2=q(x + p)^{2} = q, which also reveals the turning point of the graph.

The worked questions below are original, written from the syllabus objective. Every solution shows the full method, the points where marks are typically earned, and the slips to avoid.

Question 1

Structured Extended 4 marks

A quadratic expression is given by 2x2+5x−122x^2 + 5x - 12.

(a) Factorise 2x2+5x−122x^2 + 5x - 12 completely. [2]

(b) Hence, or otherwise, solve the equation 2x2+5x=12.2x^2 + 5x = 12. Give both roots exactly. [2]

View worked solution →

Question 2

Structured Extended 6 marks

A curve has equation y=3x2−2xy = 3x^2 - 2x and a straight line has equation y=5x+5y = 5x + 5.

The line crosses the curve at two points, PP and QQ.

(a) Show that the xx-coordinates of PP and QQ satisfy the equation 3x2−7x−5=0.3x^2 - 7x - 5 = 0. [2]

(b) This equation does not factorise. Use the quadratic formula to find the xx-coordinates of PP and QQ, giving each value correct to 2 decimal places. Show the value of the discriminant and your substitution clearly. [4]

View worked solution →

Question 3

Multiple choice Extended 1 mark

By completing the square, solve the equation

x2−5x−3=0,x^2 - 5x - 3 = 0,

giving the exact values of xx.

Which option lists both roots correctly?

View worked solution →

Question 4

Structured Extended 9 marks

(a) Factorise fully 3x2+11x−203x^2 + 11x - 20, and hence solve 3x2+11x−20=03x^2 + 11x - 20 = 0. [3]

(b) Solve x2+8x−3=0x^2 + 8x - 3 = 0 by completing the square. Give your answers correct to 3 significant figures. [3]

(c) Solve 2x2−5x−6=02x^2 - 5x - 6 = 0 using the quadratic formula. Give your answers correct to 2 decimal places. [3]

View worked solution →