SOLVING QUADRATIC EQUATIONS


Note:




Solve for x in the following equation.

Problem 4.5a:text2html_wrap_inline155tex2html_wrap_inline325






Answer:text2html_wrap_inline155tex2html_wrap_inline327 are the exact answers tex2html_wrap_inline329 are approximate answers.






Solution:


Simplify the equation tex2html_wrap_inline331 by simplifying the radicals.


eqnarray53


eqnarray59


eqnarray65


eqnarray70







Method 1:text2html_wrap_inline155Factoring

The equation tex2html_wrap_inline333 is not easily factored. Therefore, we will not use this method.







Method 2:text2html_wrap_inline155Completing the square

Add 5 to both sides of the equation tex2html_wrap_inline337


eqnarray92



Divide both sides by tex2html_wrap_inline339 :


eqnarray105



Add tex2html_wrap_inline341 to both sides of the equation:


eqnarray120



Factor the left side and simplify the right side:


eqnarray132



Take the square root of both sides of the equation:


eqnarray146



Add 1 to both sides of the equation:


eqnarray157


eqnarray162



tex2html_wrap_inline343 are the exact answers tex2html_wrap_inline345 are approximate answers.


Even though the exact answers above do not look like the answers at the beginning of the problem they are equivalent because they simplify to the same approximate answers.







Method 3:text2html_wrap_inline155Quadratic Formula

The quadratic formula is tex2html_wrap_inline347


In the equation tex2html_wrap_inline349 , a is the coefficient of the tex2html_wrap_inline351 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline355 for a, tex2html_wrap_inline357 for b, and tex2html_wrap_inline359 for c in the quadratic formula and simplify.


eqnarray191


eqnarray199


eqnarray205



tex2html_wrap_inline361 are the exact answers tex2html_wrap_inline345 are approximate answers.


Even though the exact answers above do not look like the answers in the Completing the Square method or the answers at the beginning of the problem, they are equivalent because they simplify to the same approximate answers.







Method 4:text2html_wrap_inline155Graphing

Graph the equation, tex2html_wrap_inline365 (the left side of the original equation). Graph tex2html_wrap_inline367 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline369 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 tex2html_wrap_inline373 and one at tex2html_wrap_inline375 .


The answers are tex2html_wrap_inline373 and tex2html_wrap_inline381 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=3.1296792965 by substituting 3.1296792965 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 3.1296792965 for x, then x=3.1296792965 is a solution.


Check the solution x=-1.1296792965 by substituting -1.1296792965 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 -1.1296792965 for x, then x=-1.1296792965 is a solution.





The solutions to the equation tex2html_wrap_inline411 are 3.1296792965 and -1.1296792965.






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


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


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


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