SOLVING QUADRATIC EQUATIONS

Note:




Solve for x in the following equation.

Example 4 :

tex2html_wrap_inline465


Rewrite the equation in an equivalent form without denominators by multiplying both sides by 6.

eqnarray36

eqnarray49







Method 1:text2html_wrap_inline253Factoring

Since the equation is not easily factored, we will skip this method.







Method 2:text2html_wrap_inline253Completing the square

Add 5 to both sides of the equation tex2html_wrap_inline467

eqnarray68



Divide both sides by 4 :

eqnarray73



Add tex2html_wrap_inline469 both sides of the equation:

eqnarray93



Factor the left side and simplify the right side :

eqnarray106



Take the square root of both sides of the equation :

eqnarray115



Add tex2html_wrap_inline471 to both sides of the equation.

eqnarray125

eqnarray132







Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline473

In the equation tex2html_wrap_inline475 , a is the coefficient of the tex2html_wrap_inline477 term, b is the coefficient of the x term, and c is the constant. Substitute 4 for a, -90 for b , and -5 for c in the quadratic formula and simplify.

eqnarray158

eqnarray165







Method 4:text2html_wrap_inline253Graphing

Graph tex2html_wrap_inline487 and y=0. The graph of y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline487 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 22.5554190546 and -0.055419054595. This means that there are two real answers: x=22.5554190546 and tex2html_wrap_inline505

The answers are 22.5554190546 and -0.055419054595. 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=22.5554190546 by substituting 22.5554190546 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 x=22.5554190546 for x, then x=22.5554190546 is a solution.



Check the solution x=-0.055419054595 by substituting -0.055419054595 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 -0.055419054595 for x, then x=-0.055419054595 is a solution.





The solutions to the equation tex2html_wrap_inline465 are 22.5554190546 and - 0.055419054595.





Comment:text2html_wrap_inline253You can use the exact solutions to factor the original equation.

For example, since tex2html_wrap_inline549 , then

eqnarray245


Since tex2html_wrap_inline555 , then

eqnarray256


Since the product

eqnarray263


Then tex2html_wrap_inline561 and tex2html_wrap_inline563 are factors of tex2html_wrap_inline567 .


But not the only factors.

eqnarray294


eqnarray313


eqnarray340



eqnarray360


If we multiply both sides by tex2html_wrap_inline571 , we will get :

eqnarray370



Therefore tex2html_wrap_inline465 is factored as tex2html_wrap_inline577

This means that tex2html_wrap_inline561 and tex2html_wrap_inline563 are factors of tex2html_wrap_inline585








If you would like to test yourself by working some problems similar to this example, click on Problem


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Author:Nancy Marcus

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