SOLVING QUADRATIC EQUATIONS


Note:




Solve for x in the following equation.


Problem 4.5d: tex2html_wrap_inline155 tex2html_wrap_inline429





Answer: tex2html_wrap_inline155 tex2html_wrap_inline431 are the exact answers using the Quadratic Formula.


tex2html_wrap_inline435 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_inline437 and tex2html_wrap_inline439





Solution:


Simplify the equation tex2html_wrap_inline441 by simplifying the radicals. eqnarray58





Method 1: tex2html_wrap_inline155 Factoring


The equation tex2html_wrap_inline443 is not easily factored. Therefore, we will not use this method.





Method 2: tex2html_wrap_inline155 Completing the square


Add tex2html_wrap_inline445 to both sides of the equation tex2html_wrap_inline447


eqnarray106


Divide both sides by tex2html_wrap_inline449 :


eqnarray115


Add tex2html_wrap_inline451 to both sides of the equation:


eqnarray144


Factor the left side and simplify the right side :


eqnarray157


Take the square root of both sides of the equation :


eqnarray174


Subtract tex2html_wrap_inline453 from both sides of the equation:


eqnarray184


tex2html_wrap_inline455 are the exact answers tex2html_wrap_inline457 are approximate answers.






Method 3: tex2html_wrap_inline155 Quadratic Formula


The quadratic formula is tex2html_wrap_inline459


In the equation tex2html_wrap_inline461 , a is the coefficient of the tex2html_wrap_inline463 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline467 for a, tex2html_wrap_inline469 for b, and -tex2html_wrap_inline471 for c in the quadratic formula and simplify.


eqnarray225


eqnarray225


tex2html_wrap_inline431 are the exact answers tex2html_wrap_inline457 are approximate answers.





Method 4: tex2html_wrap_inline155 Graphing


Graph the equation, tex2html_wrap_inline477 (the left side of the original equation). Graph tex2html_wrap_inline479 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline481 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 0.12530322634 and one at -0.1535874976.


The answers are tex2html_wrap_inline491 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=0.12530322634 by substituting 0.125303226341 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=-0.1535874976 by substituting -0.1535874976 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 -0.1535874976 for x, then x=-0.1535874976 is a solution.





The solutions to the equation tex2html_wrap_inline521 are 0.12530322634 and - 0.1535874976.






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