SOLVING QUADRATIC EQUATIONS

Note:




Solve for x in the following equation.


Problem 4.3b :text2html_wrap_inline253tex2html_wrap_inline300





Answer:text2html_wrap_inline253tex2html_wrap_inline302





Solution:

The equation is already set to zero.







Method 1:text2html_wrap_inline253Factoring

The equation tex2html_wrap_inline304 can be written as

z eqnarray42



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

eqnarray47







Method 2:text2html_wrap_inline253Completing the square

Add 60 to both sides of the equation tex2html_wrap_inline308


eqnarray68



Divide both sides by 2:

eqnarray83



Add tex2html_wrap_inline310 tex2html_wrap_inline312 to both sides of the equation:

eqnarray105



Factor the left side and simplify the right side :

eqnarray122



Take the square root of both sides of the equation :

eqnarray136



Subtract tex2html_wrap_inline314 from both sides of the equation :

eqnarray153


eqnarray161






Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline316

In the equation tex2html_wrap_inline300 , a is the coefficient of the tex2html_wrap_inline320 term, b is the coefficient of the x term, and c is the constant. Substitute 2 for a, 7 for b, and -60 for c in the quadratic formula and simplify.




eqnarray200


eqnarray205


eqnarray210





Method 4:text2html_wrap_inline253Graphing

Graph tex2html_wrap_inline330 (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_inline330 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 4 and -7.5. This means that there are two real answers: x=4 and tex2html_wrap_inline344

The answers are 4 and tex2html_wrap_inline348 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=4 by substituting 4 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 4 for x, then x=4 is a solution.




Check the solution tex2html_wrap_inline370 by substituting tex2html_wrap_inline372 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 tex2html_wrap_inline372 for x, then tex2html_wrap_inline370 is a solution.





The solutions to the equation tex2html_wrap_inline300 are 4 and tex2html_wrap_inline348






If you would like to review the solution to 4.3c, click on Problem


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Author:Nancy Marcus

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