,
Algebra : Quadratic Equations
 

[introduction][factorising]

[ completing the square ][using the formula]

 

 

 

Introduction

The general form of a quadratic equation is:

ax2 +bx + c

where a, b & c are constants

The expression b2 - 4ac 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

 

discriminant conditions

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

solve: 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 : 12 ...gives 13, 11

2 : 6 .....gives 8, 4

3 : 4 .....gives 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.


therefore

quadratic eq. #3

Now to solve the equation quadratic eq. #1.

factorising, as above

quadratic eq.#4

either

quad #5

or

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.
e.g.

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

 

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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