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Free Homework Math Help: Quadratic Formula

 

Quadratic Formula

The quadratic formula is considered separate from the general quadratic equation. The quadratic formula is actually the solution to the quadratic equation.

It is formally defined as:

The quadratic formula:

                  x = -b ± √b2 - 4ac
2a

is the solution of the quadratic equation ax2 + bx + c = 0 when a 0.

Roots

The quadratic formula normally has two roots whether real or complex. If the quadratic equation is a perfect square then there will be one root of multiplicity 2.

The Quadratic Formula can have at most two roots The roots of the Quadratic Formula can be real or complex

Discriminant

Probably the most important part of the quadratic formula is the discriminant. This is the part under the radical, b2 - 4ac. The discriminant determines whether the solution will have one or more roots and if they are real or complex. There are three conditions that tell what kind of answers that will result:

1. b2 - 4ac > 0         The result will have two real numbers.
2. b2 - 4ac = 0         The result will have one real number.
3. b2 - 4ac < 0         The result will have two complex numbers.

The complex answers received from the quadratic equation are also called conjugates. The forms they are in are in are a+bi and a-bi. a is the real part while bi is the imaginary part. Also, if the disciminant is a perfect square there will be rational answers.

Derivation of the Quadratic Formula

The quadratic formula is derived by completing the square on the quadratic equation ax2 + bx + c = 0.

First subtract c from both sides to get the answer:         ax2 + bx = -c
Then divide both sides by a to get:        
x2   +   b
a		x   =   -c
a
Then complete the square by adding (b
2a		)2 to both sides:
       
x2   +   b
a		x   +   (b
2a		)2   =   (b
2a		)2   -   c
a
Factor the result:        
(x   +   b
2a		)2   =   b2
4a2		   -   c
a
Match denominators:        
(x   +   b
2a		)2   =   b2
4a2		   -   4ac
4a2
Simplify:        
(x   +   b
2a		)2   =   b2-4ac
4a2
Take the square root of both sides:        
√
(x   +   b
2a		)2
   =   √b2-4ac
4a2
The 4a2 comes out of the radical as 2a with the result being:        
x   +   b
2a		   =   ±√b2-4ac
2a
Subtract the term b/(2a) from both sides:        
x   =   -b
2a		   ±   √b2-4ac
2a
       
The final result is:        
x = -b ± √b2 - 4ac
2a