SOLVING QUADRATIC EQUATIONS


Note:


Solve for x in the following equation.


Problem 4.4d: tex2html_wrap_inline155 tex2html_wrap_inline423

Answer: tex2html_wrap_inline155 tex2html_wrap_inline425





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 210. Every denominator in the original fraction will divide evenly into 210


eqnarray61


eqnarray71


eqnarray80







Method 1: tex2html_wrap_inline155 Factoring


The left side of the equation tex2html_wrap_inline427 can re rewritten in the equivalent factored form of


eqnarray93


eqnarray98



The answers are tex2html_wrap_inline429







Method 2: tex2html_wrap_inline155 Completing the square


Add 35 to both sides of the equation tex2html_wrap_inline431


eqnarray123



Divide both sides by 144 : displaymath419



Simplify : displaymath420



Add tex2html_wrap_inline433 to both sides of the equation:


eqnarray162



Factor the left side and simplify the right side:


eqnarray178



Take the square root of both sides of the equation:


eqnarray191



Subtract tex2html_wrap_inline435 from both sides of the equation:


eqnarray206


eqnarray213



The answers are tex2html_wrap_inline437







Method 3: tex2html_wrap_inline155 Quadratic Formula


The quadratic formula is tex2html_wrap_inline439


In the equation tex2html_wrap_inline441, a is the coefficient of the tex2html_wrap_inline443 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline447 for a, tex2html_wrap_inline449 for b, and tex2html_wrap_inline451 for c in the quadratic formula and simplify.



eqnarray249


eqnarray254


eqnarray259


eqnarray264



The answers are tex2html_wrap_inline437







Method 4: tex2html_wrap_inline155 Graphing


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 is located at tex2html_wrap_inline463 and the other is located at tex2html_wrap_inline465


The answers are tex2html_wrap_inline467 and tex2html_wrap_inline469 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_inline467 by substituting tex2html_wrap_inline477 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_inline477 for x, then tex2html_wrap_inline467 is a solution.




Check the solution tex2html_wrap_inline487 by substituting tex2html_wrap_inline489 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_inline489 for x, then tex2html_wrap_inline487 is a solution.







The solutions to the equation tex2html_wrap_inline155 tex2html_wrap_inline499 tex2html_wrap_inline155 are tex2html_wrap_inline155 tex2html_wrap_inline501 tex2html_wrap_inline503








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Author: Nancy Marcus

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