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.

 The Quadratic FormulaThe solution to           ax2 + bx + c  =  0is

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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