SOLVING QUADRATIC EQUATIONS



Note:



Solve for x in the following equation.

Problem 4.4c:text2html_wrap_inline253tex2html_wrap_inline515



Answer:text2html_wrap_inline253tex2html_wrap_inline517



Solution:


The equation is already equal to zero.


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 8.



eqnarray53


eqnarray61

eqnarray68






Method 1:text2html_wrap_inline253Factoring

The equation tex2html_wrap_inline519 is not easily factored. Therefore, we will not use this method.






Method 2:text2html_wrap_inline253Completing the square


Subtract 192 from both sides of the equation tex2html_wrap_inline523


eqnarray84



Add tex2html_wrap_inline525 to both sides of the equation:


eqnarray96



Factor the left side and simplify the right side:


eqnarray104



Take the square root of both sides of the equation:


eqnarray114



Subtract 7 from both sides of the equation:


eqnarray128


eqnarray132



The answers are tex2html_wrap_inline527 and tex2html_wrap_inline529







Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline531


In the equation tex2html_wrap_inline533 ,a is the coefficient of the tex2html_wrap_inline535 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline539 for a, tex2html_wrap_inline541 for b, and tex2html_wrap_inline543 for c in the quadratic formula and simplify.




eqnarray156


eqnarray161


eqnarray166


eqnarray171


eqnarray176




The answers are tex2html_wrap_inline527 and tex2html_wrap_inline529







Method 4:text2html_wrap_inline253Graphing


Graph the equation, tex2html_wrap_inline549 (the left side of the original equation). Graph tex2html_wrap_inline551 (the right side of the original equation and the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline553 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 no x-intercepts. This means that there are no real answers; the answers are imaginary.


The answers are tex2html_wrap_inline559 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 tex2html_wrap_inline565 by substituting tex2html_wrap_inline567 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 tex2html_wrap_inline567 for x, then tex2html_wrap_inline565 is a solution.



Check the solution tex2html_wrap_inline577 by substituting tex2html_wrap_inline577 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 tex2html_wrap_inline585 for x, then tex2html_wrap_inline577 is a solution.






The solutions to the equation tex2html_wrap_inline589 are tex2html_wrap_inline591







Comment:text2html_wrap_inline253You can use the exact solutions to factor the left side of the original equation minus the right side of the original equation:


eqnarray323



Since tex2html_wrap_inline593



Since tex2html_wrap_inline595



The product tex2html_wrap_inline597



Since tex2html_wrap_inline599 and tex2html_wrap_inline601


then we could say


eqnarray373



However the product of the first terms of the factors does not equal tex2html_wrap_inline603



Multiply tex2html_wrap_inline605 by tex2html_wrap_inline607



Let tex2html_wrap_inline609 s check to see if tex2html_wrap_inline611



eqnarray424


eqnarray431


eqnarray441


eqnarray452


eqnarray460



The factors of tex2html_wrap_inline613 are tex2html_wrap_inline607 , tex2html_wrap_inline617 and tex2html_wrap_inline619








If you would like to review the solution to problem 4.4d, click on Problem


If you would like to test yourself by working some problems similar to this example, click on Problem


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



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Author: Nancy Marcus

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