SOLVING QUADRATIC EQUATIONS



Note:




Solve for x in the following equation.

Problem 4.2d:tex2html_wrap_inline155 tex2html_wrap_inline313





Answer: tex2html_wrap_inline155 tex2html_wrap_inline315

tex2html_wrap_inline317 approximate solutions





Solution: tex2html_wrap_inline155


Remove the denominators from the original equation by multiplying both sides by 144.

eqnarray53


eqnarray61


eqnarray69




Set the equation tex2html_wrap_inline319 equal to zero by subtracting 2016x and 1584 from both sides of the equation.


eqnarray76


eqnarray80


eqnarray84







Method 1:Factoring



The equation tex2html_wrap_inline323 is not easily factored so we will skip this method.






Method 2:Completing the square


Subtract 18 from both sides of the equation tex2html_wrap_inline325 .



eqnarray99


Add 1575 to both sides of the equation : tex2html_wrap_inline327



Divide both sides by 144: tex2html_wrap_inline329



eqnarray115



Add tex2html_wrap_inline331 to both sides of the equation:



eqnarray131



Factor the left side and simplify the right side :

eqnarray143



Take the square root of both sides of the equation :

eqnarray151



Add tex2html_wrap_inline333 from both sides of the equation :

eqnarray160








Method 3:Quadratic Formula



The quadratic formula is tex2html_wrap_inline335



In the equation tex2html_wrap_inline325 , a is the coefficient of the tex2html_wrap_inline339 term, b is the coefficient of the x term, and c is the constant. Simply insert 144 for a, -1936 for b, and -1575 for c in the quadratic formula and simplify.



eqnarray189



eqnarray196



eqnarray207








Method 4:Graphing

Graph tex2html_wrap_inline343 (formed by subtracting the right side of the original equation from the left side of the original equation. Graph y=0 (the x-axis).

What you will be looking for is where the graph of tex2html_wrap_inline347 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 14.2139358 and -0.769491307. This means that there are two real answers: x = 14.2139358 and -0.769491307.




The answers are x = 14.2139358 and - 0.769491307. 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=14.2139358 by substituting 14.2139358 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 14.2139358 for x, then x=14.2139358 is a solution.



Check the solution x=-0.769491307 by substituting -0.769491307 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 -0.769491307 for x, then x=-0.769491307 is a solution.



The solutions to the equation tex2html_wrap_inline385 are 14.2139358 and tex2html_wrap_inline389




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Author: Nancy Marcus

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