PURE MATHEMATICS - Algebra

 

Quadratic Equations

 

 

first principles

notation

the formula

 

 

 

Introduction

 

The general form of a quadratic equation is:

 

 

where a, b & c are constants

 

 

The expression is called the discriminant and given the letter Δ (delta).

 

All quadratic equations have two roots/solutions.

 

These roots are either REAL, EQUAL or COMPLEX*.

 

*complex - involving the square root of -1

 

 

discriminant graph

 

 

 

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Solution by Factorising - This is best understood with an example.

 

 

Example

 

Solve:      

 

 

You must first ask yourself which two factors when multiplied will give 12 ?

 

The factor pairs of 12 are :   1 x 12,   2 x 6   and   3 x 4

 

You must decide which of these factor pairs added or subtracted will give 7 ?

 

 

1 : 12 ...gives 13, 11
2 : 6 .....gives 8, 4
3 : 4 .....gives 7, 1

 

 

 

 

Which combination when multiplied makes +12 and when added gives -7 ?


These are the choices:

 

 

(+3)(+4)
(-3)(+4)
(+3)(-4)
(-3)(-4)

 


Clearly, (-3)(-4) are the two factors we want.


therefore,

 

 

 

Now to solve the equation,

 

factorising, as above putting the function equal to zero,

 

 

either,

or

 

for the equation to be true.

 

So the roots of the equation are:            

 

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Completing the Square

 

This can be fraught with difficulty, especially if you only half understand what you are doing.

 

The method is to move the last term of the quadratic over to the right hand side of the equation and to add a number to both sides.

 

The reason for doing this is so the left hand side can be factorised as the square of two terms.


example

 

     

     

   

         

            

               

 

 

 

   

 

 

However, there is a much neater way of solving this type of problem.

 

That is by remembering to put the equation in the following form:

 

 

 

                                                   

 

 

using the previous example,

 

completing the square#3

 

 

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Using the Formula

 

The two solutions of quadratic equations in the form:

 

quadratic equation form

 

are given by the formula:

the quadratic equation formula

 

 

Proof

 

The proof of the formula is by using 'completing the square'.

 

 

proof of the equation for solving quadratic equations

 

 

Example  

 

Find the two values of x that satisfy the following quadratic equation:

 

example#1 - the formula

 

 

 

example #1 using the formula

 

 

 

 

 

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