SOLVING QUADRATIC EQUATIONS



Note:




Solve for x in the following equation.

Problem 4.4b:text2html_wrap_inline253 tex2html_wrap_inline411




Answer:text2html_wrap_inline253tex2html_wrap_inline413




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 20.


eqnarray60


eqnarray70


eqnarray79







Method 1:text2html_wrap_inline253Factoring

The equation tex2html_wrap_inline415 can be rewritten in the equivalent factored form of tex2html_wrap_inline417


eqnarray86


The answers are tex2html_wrap_inline419 and tex2html_wrap_inline421 using the factoring method.






Method 2:text2html_wrap_inline253Completing the square


Subtract 4 from both sides of the equation tex2html_wrap_inline425


eqnarray114



Divide both sides by 45:


eqnarray122



Simplify:


eqnarray138



Add tex2html_wrap_inline427 to both sides of the equation:


eqnarray156



Factor the left side and simplify the right side:


eqnarray169



Take the square root of both sides of the equation:


eqnarray180



Add tex2html_wrap_inline429 to both sides of the equation:


eqnarray192


eqnarray199



The answers are tex2html_wrap_inline419 and tex2html_wrap_inline421 using the method of Completing the Square.






Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline435

In the equation tex2html_wrap_inline437 ,a is the coefficient of the tex2html_wrap_inline439 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline443 for a, tex2html_wrap_inline445 for b, and tex2html_wrap_inline447 for c in the quadratic formula and simplify.




eqnarray239


eqnarray244


eqnarray249


eqnarray254


eqnarray263




The answers are tex2html_wrap_inline419 and tex2html_wrap_inline421 using the method of the Quadratic Fromula.






Method 4:text2html_wrap_inline253Graphing

Graph the equation, tex2html_wrap_inline455 (the left side of the original equation). Graph tex2html_wrap_inline457 (the right side of the original equation and the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline459 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 0.3333333 and one at 0.266667.


The answers are tex2html_wrap_inline419 and tex2html_wrap_inline471 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_inline477 by substituting tex2html_wrap_inline419 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_inline419 for x, then tex2html_wrap_inline477 is a solution.



Check the solution tex2html_wrap_inline489 by substituting tex2html_wrap_inline421 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_inline421 for x, then tex2html_wrap_inline489 is a solution.






The solutions to the equation tex2html_wrap_inline155 tex2html_wrap_inline501 tex2html_wrap_inline155 are tex2html_wrap_inline155 tex2html_wrap_inline419 tex2html_wrap_inline155 and tex2html_wrap_inline155 tex2html_wrap_inline505








If you would like to review the solution to problem 4.4c, 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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