SOLVING QUADRATIC EQUATIONS

Note:




Solve for x in the following equation.

Problem 4.5b:text2html_wrap_inline155tex2html_wrap_inline377






Answer:text2html_wrap_inline155tex2html_wrap_inline379 are the exact answers using the Quadratic Formula.


tex2html_wrap_inline383 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_inline385 and tex2html_wrap_inline387






Solution:


Simplify the equation tex2html_wrap_inline389 by simplifying the radicals.


eqnarray55



Eliminate the denominator 5 by multiplying both sides by 5.


eqnarray68







Method 1:text2html_wrap_inline155Factoring

The equation tex2html_wrap_inline395 is not easily factored. Therefore, we will not use this method.







Method 2:text2html_wrap_inline155Completing the square

Subtract 1 to both sides of the equation tex2html_wrap_inline399


eqnarray98



Divide both sides by tex2html_wrap_inline401


eqnarray106



Add tex2html_wrap_inline403 to both sides of the equation:


eqnarray128



Factor the left side and simplify the right side :


eqnarray141



Take the square root of both sides of the equation :


eqnarray150



Add tex2html_wrap_inline405 to both sides of the equation :


eqnarray160



tex2html_wrap_inline407 are the exact answers tex2html_wrap_inline409 are approximate answers.







Method 3:text2html_wrap_inline155Quadratic Formula

The quadratic formula is tex2html_wrap_inline411


In the equation tex2html_wrap_inline413 , a is the coefficient of the tex2html_wrap_inline415 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline419 for a, tex2html_wrap_inline421 for b, and tex2html_wrap_inline423 for c in the quadratic formula and simplify.


eqnarray199


eqnarray210



tex2html_wrap_inline425 are the exact answers tex2html_wrap_inline427 are approximate answers.







Method 4:text2html_wrap_inline155Graphing

Graph the equation, tex2html_wrap_inline429 (the left side of the original equation). Graph tex2html_wrap_inline431 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline433 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 -0.02354671551 and one at -1.20119815588.


The answers are -0.02354671551 and -1.20119815588. 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=-0.02354671551 by substituting -0.02354671551 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 -0.02354671551 for x, then x=-0.02354671551 is a solution.


Check the solution x=-1.20119815588 by substituting -1.20119815588 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 -1.20119815588 for x, then x=-1.20119815588 is a solution.





The solutions to the equation tex2html_wrap_inline475 are-0.02354671551 and-1.20119815588.






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


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


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