Mathematics

 

Quadratic Equations

In this tutorial, you will learn:

  • What a quadratic equation is
  • How to solve by factorisation
  • How to solve by quadratic formula
  • How to solve quadratic equations by completing the square
  • How to solve a quadratic equation graphically
  • How to generate quadratic equations randomly by using a programme, at the end of the tutorial for practice
  • How to use Microsoft Excel in solving quadratic equations by using the formula method

Quadratic Equations

An equation in the form of ax2 + bx + c = 0 is called a quadratic equation.

E.g.

A quadratic equation has two solutions; that means there are two values for x that satisfy the equation. There are four different ways to solve a quadratic equation:


  1. Factorizing Method
  2. Formula Method
  3. Graphical Method
  4. Completing the Square Method

Factorizing

E.g.1
x2 + 8x = 0
x(x + 8) = 0
x = 0 or (x + 8) = 0
x = 0 or x = -8

E.g.2

x2= 6x
x2 - 6x = 0
x(x - 6) = 0
x = 0 or (x - 6) = 0
x = 0 or x = 6

E.g.3

x2 + 6x + 8 = 0
x2 + 4x + 2x + 8 = 0
x(x + 4) + 2(x + 4) = 0
(x + 4)(x + 2) = 0
(x + 4) = 0 or (x + 2) = 0
x = -4 or x = -2

E.g.4

x2 - 6x + 8 = 0
x2 - 4x - 2x + 8 = 0
x(x - 4) - 2(x - 4) = 0
(x - 4) = 0 or (x - 2) = 0
x = 4 or x = 2<

E.g.5

x2 + 6x - 16 = 0
x2 + 8x - 2x - 16 = 0
x(x + 8) - 2(x + 8) = 0
(x + 8) = 0 or (x - 2) = 0
x = -8 or x = 2

E.g.6

2x2 + 13x + 6 = 0
2x2 + 12x + x + 6 = 0
2x(x + 6) + 1(x + 6) = 0
(x + 6) = 0 or (2x + 1) = 0
x = -6 or 2x = -1
x = -6 or x = -1/2

E.g.7

x2 - 9/4 = 0
(x + 3/2)(x - 3/2) = 0
x + 3/2 = 0 or x - 3/2 = 0
x = -3/2 or x = 3/2

Formula Method

If ax2 + bx + c = 0, then
x = [-b ±√(b2 - 4ac) ]/ 2a


E.g.1

x2 - 6x + 8 = 0
a = 1; b = -6; c = 8
x = -(-6) ±√((-6)2 - 4(1)(8)) / 2(1)
x = 6 ±√(36 - 32) / 2
x = 6 ±√(4) / 2
x = (6 ± 2 )/ 2
x = 4 or x = 2

E.g.1

2x2 - 5x + 3 = 0
a = 2; b = -5; c = 3
x = -(-5) ±√((-5)2 - 4(2)(3)) / 2(2)
x = 5 ±√(25 - 24) / 4
x = 5 ±√(1) / 4
x = (6 ± 1 )/ 4
x = 1.5 or x = 1

 

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Graphical Method

In this method, a graph is plotted for a quadratic function. The graph takes the typical shape, known as parabola.

E.g. Solve x2 + 5x - 7 = 0

First of all, make a table for both x and y of the function.

x y
-2 -13
-1 -11
0 -7
1 -1
2 7
3 17
quadratic graph

 

Now, plot a graph of y against x. Note the points at which the curve the crosses the x-axis. They are the solutions of the quadratic function. The solutions are:
x = 1.1 or x = -6.1

The following animation is interactive: it shows how to solve a quadratic equation by a graph; by clicking on the button, you can generate a random equation and its solutions appear at the same time. If there are no solutions - the graph being above the x-axis - instead of solutions, the word, undefined, appears in those places.

 

 

Completing the Square Method

x2 + 4x - 5 = 0
Let (x + a)2 + b = x2 + 4x - 5
x2 + 2ax + a2 + b = x2 + 4x - 5
Now, make the coefficients of x and the constant equal.
x => 2a = 4
a =2
a2 + b = -5
4 + b = -5
b = -9
(x + 2)2 - 9 = 0
(x +2)2 = 9
(x + 2) = ±3
x = -2 ±3
x = 1 or -5


 

Now, in order to complement what you have just learnt, work out the following questions:

Click the button to get the quadratic equations; solve them by all four methods to master the techniques.

Quadratic Equation Generator

A random equation for each click will appear here.

 

 

 

 

Recommended Reading


 

Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7th edition in print.

Recommended - GCSE & iGCSE


 

This is the best book available for the new GCSE(9-1) specification and iGCSE: there are plenty of worked examples; a really good collection of problems for practising; every single topic is adequately covered; the topics are organized in a logical order.

Recommended for A Level


 

This is the best book that can be recommended for the new A Level - Edexcel board: it covers every single topic in detail;lots of worked examples; ample problems for practising; beautifully and clearly presented.