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.
  • x2 + 6x + 8 = 0
  • 2x2 - 5x + 6 = 0
  • x2 - 6 = 0
  • x2 - 6x = 0

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

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

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


Solving Quadratic Equations with MS Excel

The following animation shows how MS Excel can be easily programmed to solve quadratic equations. It is done by VBA code, which is fairly simple to grasp.

quadratic solver on excel

 

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.

 

 

 

 

 

 

Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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Recommended Reading

 

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