Solving Quadratic Equations with the Quadratic Formula
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Quadratic Formula:
For ,

The solutions of some quadratic equations, (), are not rational, and cannot be obtained by factoring.  For such equations, the most common method of solution is the quadratic formula. 

Note:  The quadratic formula can be used to solve ANY quadratic equation, even those that can be factored.  Be sure you know this very useful formula!!!


Examples:

1. 

By factoring
(this equation is factorable):
By Quadratic Formula:  a = 1,  b = 2,  c = -8

Hints:
Be careful with the signs of the values a, b and c.  Don't drop the sign when substituting into the formula.

Also remember your rules for multiplying and adding signed numbers as you solve the formula.

 

2. 

This equation cannot be solved by factoring.
By Quadratic Formula:  a = 3,  b = -10,  c = 5
Hints: 
Notice how the value for b was substituted into the formula using parentheses (-10).   This helps you to remember to deal with the negative value of b.

Also, notice how the (-10)2 is actually a positive value.  When you square a value, the answer is always positive.

If needed, these answers can be estimated as decimal values, such as (rounded to 3 decimal places):
x = 2.721;        x = 0.613
The radical answers are the "exact" answers.
The decimal answers are "approximate" answers.

 

3. 

This equation cannot be solved by factoring.
By Quadratic Formula:  a = 1,  b = 4,  c = 5
Hints:
Remember that a negative value under the radical is creating an imaginary number (a number with an i).
 

 

4. 

By factoring
(this equation is factorable):
By Quadratic Formula:  a = 1,  b = -4,  c = 4
Hints:
When the value under the radical turns to zero, there will appear to be only one answer to the problem (since the plus/minus option is gone).  This really means that the one root is repeating itself, as seen in the factoring solution.

 

5. 

Whoa!!  Stop the presses!!!
This problem cannot be solved using the Quadratic Formula
until it is set equal to zero.

By Quadratic Formula:  a = 2,  b = 1,  c = -1/2

Hints:




 


Deriving the Quadratic Formula:

The quadratic formula is derived from the quadratic equation by a process called "completing the square".  Here is how it was developed:

The process used here is explained in more detail under completing the square .

Isolate the variable terms.  Move the constant, c,  to the right.

Prepare to form a perfect square on the left.

To obtain the constant value needed for the perfect square, take half of the coefficient of the x-term and square it.  Add this value to both sides of the equation.

Find a common denominator on the right.

Combine terms on the right.

Write the perfect square on the left.

Take the square root of both sides.

Simplify the denominator from the square root.

Subtract the constant away from the x.

Finally, the quadratic formula appears.  Whew!!!