

A quadratic equation is a polynomial equation of degree two,
which can be written in the form ax^{2} + bx + c = 0, where x is a variable and a, b and c are constants with a ≠ 0. 
Why does "quadratic" refer to equations of degree two?
While the prefix "quadri" (in Latin) means four, the word "quadrus" means a square (which has four sides). The word "quadratus" means "squared".
In early mathematics, quadratic equations were used in connection with geometric problems involving squares. When you find the area of a square, for example, the area is expressed in "square" units (with the units being raised to the second power). Raising to a power of two is referred to as "squaring", and equations with a power of two are called quadratic equations.


Solving quadratic equations:
Solving a quadratic equation can range from being a simple task, to being a challenge. In this quadratic section, we will be building our arsenal of strategies for solving quadratic equations. 
Quadratic Equation Strategies:


To solve a quadratic equation by factoring:
For more details on the process of factoring, see the Factoring section.
The information presented on this page is meant as a refresher of those factoring skills, as they pertain to quadratic equations.

Factoring Method 
1. 
Express the equation in the form ax^{2} + bx + c = 0. 
2. 
Factor the left hand side (if 0 is on the right). 
3. 
Set each of the two factors equal to zero. 
4. 
Solve for x to determine the roots (or zeros). 

Simple quadratic equations with rational roots can be solved by factoring. Let's refresh our memories on factoring these simple quadratic equations as they appear in different situations.
Refer to Factoring for more examples.

Examples of Solving Quadratic Equations by Factoring:
Factoring with GCF
(greatest common factor):

Solve: 4x^{2}  28x = 0
4x(x  7) = 0
4x = 0; x  7 = 0
x = 0; x = 7


Find the largest value which can be factored from each term on the left side of the quadratic equation.
The roots (zeros) correspond to the locations of the xintercepts of the function y = 4x^{2}  28x. 
Factoring Trinomial with Leading Coefficient of One:

Solve: x^{2} + 2x  15 = 0
(x + 5)(x  3) = 0
x + 5 = 0; x  3 = 0
x = 5; x = 3


When the leading coefficient is one, the product of the roots will be the constant term, and the sum of the roots will be the coefficient of the middle xterm. 
Factoring Difference of Two Squares:

Solve: x^{2}  81 = 0
(x + 9)(x  9) = 0
x + 9 = 0; x  9 = 0
x = 9; x = 9


Remember the pattern for the difference of two squares, where the factors are identical except for the sign between the terms.

Factoring Trinomial with Leading Coefficient Not One:

Solve: 3x^{2} + 11x  4 = 0
(3x  1)(x + 4) = 0
3x  1 = 0; x + 4 = 0
x = 1/3; x = 4


Life gets more difficult when the leading coefficient is not one. Check out the Factoring section to see more strategies for factoring these pesky problems. 
Where's the x^{2 }?

Solve: 2x(x + 4) = x  3
2x^{2 }+ 8x = x  3
2x^{2 }+ 7x + 3 = 0
(2x + 1)(x + 3) = 0
2x + 1 = 0; x + 3 = 0
x =  1/2; x = 3


Sometimes you have to "work" on the equation to get the needed quadratic form. In this case, distribute, and the x^{2} will appear. 
Dealing with Proportions:

Solve:
2( x + 4) = ( x + 1)( x  2)
2 x + 8 = x^{2 }  x  2
0 = x^{2 }  3 x  10
0 = ( x  5)( x + 2)
x  5 = 0; x + 2 = 0
x = 5; x = 2


x^{2} may appear when cross multiplying ("product of the means equals product of the extremes") is employed in a proportion. 

Square
Root Method (ax^{2} is the only variable term) 
1. 
Isolate the ax^{2} term on one side of the equation. 
2. 
Take the square root of both sides. 
3. 
Remember to use ±, as there are two solutions. 
4. 
Express the roots (or zeros). 

The square root method applies only to a one specific situation. This method works when there is no middle bxterm in the equation. The only variable in the equation is an x^{2}term.

Examples of Solving Quadratic Equations by Square Root Method:
Easiest SetUp (has "x^{2} = __"): 
Solve: x^{2} = 121
x = ±11


We saw this example under the previous method of factoring. Here it is again, being solved by the square root method. Notice the answers are the same.

Difference of Two Squares:

Solve: x^{2}  81 = 0
x^{2} = 81
x = ± 9


We saw this example under the previous method of factoring. Here it is again, being solved by the square root method. Notice the answers are the same.

Only variable is the x^{2} term, with subtraction of constant. 
Solve: x^{2}  12 = 0
x^{2} = 12


This example shows the numeric value (the constant), subtracted from the x^{2} term.
Approximate decimal answers:
x = 3.464102 and
x = 3.4641016 
Only variable is the x^{2} term, with addition of constant. 
Solve: x^{2} + 9 = 0
x^{2} = 9


This example shows the numeric value (the constant), added to the x^{2} term, which creates "complex" answers containing the imaginary "i". 
There are tools more powerful than factoring and square root method
for solving quadratic equations.
Check out these other strategies for solving quadratic equations:
completing the square and quadratic formula. 

For calculator help with graphing parabolas
click here. 


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