Quadratic equations have two roots {a₁, a₂} that are sometimes referred to as solutions.  Because there are two roots one may write the quadratic as a product of two first ordered polynomials: 

f(x) = a₀(x-a₁)(x-a₁)= a₀x₂-a₀(a₁+a₂)x+ a₀a₁a₂ [1] 

This general form usually is written as 

i)        [2.1]

ii)     [2.2]

     To solve quadratic equations of the form  we refer to the general form shown above.  It is seen that the two roots are can be extracted from the terms 2a₀(a₁+a₂) and a₀a₁a₂.  A little thought shows that: (20(a₁+a₂))2 minus 4a₀(a₀a₁) equals (4a₀)2(a₁+a₂)2. When the square root of this expression is either added or subtracted from  (-1)(-a₀(a₁+a₂))  and the result divided by 2a₀,  either a₁ or a₂ is returned.

 This procedure is normally written in a more compact form, applicable for equation 2.1, and is known as the Quadratic Formula.

1)         [3.1]

Note that the coefficient of the term x₀ contains the product of the two roots.  When this term is missing, at least one of the roots is missing, resulting in the equation: 

 [4.1]  which can be factored: ax(x+(b/a))=0 whose roots are 0 and –b/a respectively. 

Given that c = a₀a₁a₂ and  = 2a₀(a₁ + (a₁-a₂), one can deduce that

 is another version of the quadratic formula.  

We can predict the nature of the roots by examining the discriminant.

There are three possible combinations: 

i)                    If  then the equation has no real roots.

ii)                   If  then the equation has two equal roots.

iii)                 If  then the equation has two real and unique roots.

Quadratic equations of the form  are very useful when we want to graph an equation.  This is because a gives us the compression and orientation, and the parabola’s vertex is (p,q).  However, in order to solve them we must change them into the form .  This is done in the following manner: 

Now we can solve using the quadratic formula.

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