SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Problem 4.2a:

tex2html_wrap_inline184

Answer: x = -1, 6


Solution:

Set the equation equal to zero by subtracting 16 and adding 2x to both sides of the equation.

eqnarray22




Method 1:Factoring

The equation tex2html_wrap_inline190 can be written as

eqnarray31

The only way a product can equal zero is for aat least one of the factors to have a value of zero:

eqnarray36




Method 2:Completing the square

Add 6 to both sides of the equation tex2html_wrap_inline192

eqnarray42



Add tex2html_wrap_inline194 to both sides of the equation:

eqnarray52



Factor the left side and simplify the right side :

eqnarray60



Take the square root of both sides of the equation :

eqnarray68



Add tex2html_wrap_inline196 to both sides of the equation :

eqnarray77

and

eqnarray89





Method 3:Quadratic Formula

The quadratic formula is tex2html_wrap_inline198



In the equation tex2html_wrap_inline200 , a is the coefficient of the tex2html_wrap_inline202 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, -5 for b, and -6 for c in the quadratic formula and simplify

.

eqnarray112

eqnarray116

and

eqnarray122





Method 4:Graphing

Graph tex2html_wrap_inline208 (formed by subtracting the right side of the original equation from the left side of the original equation. Graph y=0 (the x-axis). What you will be looking for is

where the graph of tex2html_wrap_inline212 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation.





You can see from the graph that there are two x-intercepts located at 6 and -1. This means that there are two real answers: x=6 and tex2html_wrap_inline222



The answers are 6 and -1. These answers may or may not be solutions to the original equation. You must check the answers with the original equation.



Check these answers in the original equation.



Check the solution x=6 by substituting 6 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 6 for x, then x=6 is a solution.





Check the solution x=-1 by substituting -1 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -1 for x, then x=-1 is a solution.





The solutions to the equation tex2html_wrap_inline200 are -1 and 6.


If you would like to review the solution to 4.2b, click on Solution.

If you would like to go back to the problem page, click on Problem.

If you would like to go back to the equation table of contents, click on Contents.

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Author:Nancy Marcus

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