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Coursework Notes - Algebra


Quadratic Equations


solution by factorising   by completing the square using the formula



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



quadratic eq. #1


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 x 12 13, 11

2 x 6 8, 4

3 x 4 7, 1


quadratic eq. #2


Which combination when multiplied makes +12 and which 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.



quadratic eq. #3



Now to solve the equation:


quadratic eq. #1


factorising, as above


quadratic eq.#4




quad #5




quad #6


for the equation to be true.


So the roots of the equation are:           quad #7



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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 so that the left hand side can be factorised as the square of two terms.

completing the square#1



However, there is a much neater way of solving this type of problem, and that is by remembering to put the equation in the following form:


completing the square#2

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


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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