SOLVING QUADRATIC EQUATIONS



Note:




Solve for x in the following equation

Example 2: tex2html_wrap_inline155 tex2html_wrap_inline567

If you have forgotten how to manipulate fractions, click on Fractions for a review.


Remove all the fractions by writing the equation in an equivalent form without fractional coefficients. In this problem, you can do it by multiplying both sides of the equation by 12. All the denominators of the original equation divide into 12 evenly.


eqnarray41


Set the equation equal to zero by subtracting 2x from both sides of the equation:


eqnarray59







Method 1: tex2html_wrap_inline155 Factoring

The equation tex2html_wrap_inline569 is not easily factored. Therefore, we will not use this method.







Method 2: tex2html_wrap_inline155 Completing the square

Divide both sides of the equation by 9.

eqnarray71


Subtract tex2html_wrap_inline573 from both sides of the equation:


displaymath543



Simplify : tex2html_wrap_inline575



Add tex2html_wrap_inline577 to both sides of the equation :


displaymath544



Factor the left side and simplify the right side:

displaymath545



Take the square root of both sides of the equation :


displaymath546



Add tex2html_wrap_inline579 to both sides of the equation:


displaymath547


displaymath548


displaymath549







Method 3: tex2html_wrap_inline155 Quadratic Formula

The quadratic formula is tex2html_wrap_inline581

In the equation tex2html_wrap_inline583 , a is the coefficient of the tex2html_wrap_inline585 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline589 for a , tex2html_wrap_inline591 for b, and tex2html_wrap_inline593 for c in the quadratic formula and simplify.





eqnarray172


eqnarray182







Method 4: tex2html_wrap_inline155 Graphing

Graph the equation, tex2html_wrap_inline595 (formed by subtracting the right side of the equation from the left side of the equation). Graph tex2html_wrap_inline597 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline599 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, one at 9.48527114763 and one at 0.07028440793.


The answers are 9.48527114763 and tex2html_wrap_inline609 These answers may or may not be solutions to the original equations. You must verify that these answers are solutions.



Check these answers in the original equation.



Check the solution x=9.48527114763 by substituting 9.48527114763 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 9.48527114763 for x, then x=9.48527114763 is a solution.

Check the solution x=0.07028440793 by substituting 0.07028440793 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 0.07028440793 for x, then x=0.07028440793 is a solution.







The solutions to the equation tex2html_wrap_inline639 are 9.48527114763 and 0.07028440793.







Comment: tex2html_wrap_inline155 You can use the exact solutions to factor the left side of the original equation set to zero:.


eqnarray272



Since tex2html_wrap_inline645 tex2html_wrap_inline647



Since tex2html_wrap_inline649



The product tex2html_wrap_inline651



Since tex2html_wrap_inline653 and tex2html_wrap_inline655



then we could say


displaymath550



However the product of the first terms of the factors does not equal tex2html_wrap_inline657



Multiply tex2html_wrap_inline659 by tex2html_wrap_inline661


Let's check to see if tex2html_wrap_inline665


displaymath551


tex2html_wrap_inline667


displaymath552


displaymath553


displaymath554



The factors of tex2html_wrap_inline669 are tex2html_wrap_inline671 and tex2html_wrap_inline673









If you would like to work another example, click on Example


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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