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.

 

 

 

 

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.

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