The Quadratic Formula

Quadratic Formula

Lets complete the square for

        ax² + bx + c

1. a(x² + b/a x) + c
   
   
2.    -b
                                                           
      2a       
                     
   
3.     b²
               
      4a²    
   
   
   
4.               b             b²          b²          
   a( x²  +        x  +           -           )  + c                 
                  a             4a²       4a²    
   
   
   
5.                 b              b²            b²          
   a[ ( x²  +        x  +            ) -           ]  + c                 
                    a             4a²           4a²    
   
   
   
   
6.                b                b²          
   a[ ( x +          )²  -             ]  + c                 
                 2a              4a²              
   
   
   
7.              b               b²          
   a( x +          )²  -              + c                 
               2a              4a              
   

Now what if we want to solve the equation

        ax² + bx + c = 0

We can equivalently solve

                     b               b²          
        a( x +          )²  -              + c  = 0               
                    2a              4a              

by the root method

                     b               b²          
        a( x +          )²  =              -  c                 
                    2a               4a              

                   b                b²           c  
        ( x +          )²  =              -                    
                  2a               4a²          a     

              b²    -    4ac  
        =                                    
                    4a²    

Now take a square root of both sides to get

Finally subtract b/2a from both sides to get the quadratic formula.

Memorize This Formula!

Example  

Solve 

        3x² - 2x + 5

Solution

        a = 3         b = -2          c = 5

We have 

The Discriminant

We define the discriminant as

        D = b²  - 4ac

is a convenient measure of determining how many real roots (solutions) there are.  Notice that D is the expression inside of the square root sign in the quadratic formula.  Since the square root of a negative number produces only complex numbers, we see that if D is negative, then there will be no real roots.  If D is a positive number, then the quadratic formula will produce two roots (one for the plus and one for the minus).  If D is 0 then plus 0 and minus 0 are the same number, so we get only one root.  The table below summarizes.

 

D	Number of Real Roots
Positive	2
Negative	0
Zero	1

Example:

How many roots are there for the equation:

        3x² - 5x + 1  =  0

We have 

        D  =  25 - 12  >  0

hence there are two real roots.

The Sum and Product of the Roots

Since the two roots of a quadratic are

and

then if we add the two roots, we get:

and if we multiply the two roots we get

Note that if a is 1 then the sum of the roots is -b and the product is c.  This relates to factoring when we find two numbers that add to b and multiply to c.

Example

What are the sum and product of the roots of

        4x² -3x + 2

Solution 

The sum is 

            b          3
        -        =           
            a          4

and the product of the roots is 

            c           2           1
                  =          =                                      
            a           4           2

Determining the Quadratic Equation From the Roots

If we know the roots of a quadratic then it is easy to find the original quadratic by using the zero product formula in reverse.

Example

Find an equation of a quadratic that has roots 2 and -4/3.

Solution

We can write:

        (x - 2)(x - (-4/3))  =  (x - 2)(x + 4/3) 

                      4                  8                2           8
        =  x² +         x - 2x -         =  x² -       x -        
                      3                  3                3           3

Exercise:  

Find a quadratic with the following roots

1. 0 and -1/2                         
   
   

2. 3 -      and     3 +     
   
   

3. 4 + i     and     4 - i            

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