SOLVING QUADRATIC EQUATIONS


Note:




Solve for x in the following equation.


Problem 4.5c: tex2html_wrap_inline155 tex2html_wrap_inline394






Answer: tex2html_wrap_inline155 tex2html_wrap_inline396 are the exact answers using the Quadratic Formula.


tex2html_wrap_inline400 are the exact answers using the Completing the Square Method.


even though the two answers look different, they are equivalent because both yield the same approximate answers of tex2html_wrap_inline404 and tex2html_wrap_inline406






Solution:


Simplify the equation tex2html_wrap_inline394 .


Divide both sides by tex2html_wrap_inline410


eqnarray58






Method 1: tex2html_wrap_inline155 Factoring


The equation tex2html_wrap_inline412 is not easily factored. Therefore, we will not use this method.






Method 2: tex2html_wrap_inline155 Completing the square


Add tex2html_wrap_inline414 to both sides of the equation tex2html_wrap_inline416


eqnarray117


Add tex2html_wrap_inline418 to both sides of the equation:


eqnarray130


Factor the left side and simplify the right side :


eqnarray142


Take the square root of both sides of the equation :


eqnarray151


Subtract tex2html_wrap_inline420 from both sides of the equation :


eqnarray161


tex2html_wrap_inline422 are the exact answers tex2html_wrap_inline424 are approximate answers.






Method 3: tex2html_wrap_inline155 Quadratic Formula


The quadratic formula is tex2html_wrap_inline426


In the equation tex2html_wrap_inline428 ,a is the coefficient of the tex2html_wrap_inline430 term, b is the coefficient of the x term, and c is the constant. Substitute 1 for a, tex2html_wrap_inline436 for b , and tex2html_wrap_inline438 for c in the quadratic formula and simplify.


eqnarray201


tex2html_wrap_inline396 are the exact answers tex2html_wrap_inline424 are approximate answers.






Method 4: tex2html_wrap_inline155 Graphing


Graph the equation, tex2html_wrap_inline444 (the left side of the original equation). Graph tex2html_wrap_inline446 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline448 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 6.57797305561 and one at -10.31963044.


The answers are 6.57797305561 and -10.31963044. 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 = 6.57797305561 by substituting 6.57797305561 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 6.57797305561 for x, then x = 6.57797305561 is a solution.



Check the solution x = -10.31963044 by substituting -10.31963044 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 -10.31963044 for x, then x = - 10.31963044 is a solution.







The solutions to the equation tex2html_wrap_inline490 are 6.57797305561 and -10.31963044.







If you would like to review the solution to problem 4.5d, click on Problem

If you would like to go back to the beginning of the quadratic section, click on Quadratic

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


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