SOLVING QUADRATIC EQUATIONS



Note:




Solve for x in the following equation.

Problem 4.4a:text2html_wrap_inline253tex2html_wrap_inline377




Answer:text2html_wrap_inline253tex2html_wrap_inline379 are the exact answers tex2html_wrap_inline381 are approximate answers.




Solution:

Set the equation equal to zero by subtracting 2 from both sides.


eqnarray40


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 3.


eqnarray57


eqnarray63







Method 1:text2html_wrap_inline253Factoring

The equation tex2html_wrap_inline385 is not easily factored. Therefore, we will not use this method.







Method 2:text2html_wrap_inline253Completing the square

Add 36 to both sides of the equation tex2html_wrap_inline389


eqnarray83



Add tex2html_wrap_inline391 to both sides of the equation:


eqnarray97



Factor the left side and simplify the right side:


eqnarray108



Take the square root of both sides of the equation:


eqnarray117



Add 6 to both sides of the equation:


eqnarray125


eqnarray129



tex2html_wrap_inline379 are the exact answers tex2html_wrap_inline381 are approximate answers.







Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline397


In the equation tex2html_wrap_inline399 ,a is the coefficient of the tex2html_wrap_inline401 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline405 for a , tex2html_wrap_inline407 for b, and tex2html_wrap_inline409 for c in the quadratic formula and simplify.



eqnarray152


eqnarray157


eqnarray162


eqnarray167


eqnarray171



tex2html_wrap_inline379 are the exact answers tex2html_wrap_inline381 are approximate answers.







Method 4:text2html_wrap_inline253Graphing

Graph the equation, tex2html_wrap_inline415 (formed by subtracting the right side of the equation from the left side of the equation). Graph tex2html_wrap_inline417 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline419 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 14.485281 and one at -2.485281.

The answers are tex2html_wrap_inline429 and tex2html_wrap_inline431 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=14.485281 by substituting 14.485281 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 14.485281 for x, then x=14.485281 is a solution.

Check the solution x=-2.485281 by substituting -2.485281 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 -2.485281 for x, then x=-2.485281 is a solution.







The solutions to the equation tex2html_wrap_inline461 are tex2html_wrap_inline463 and tex2html_wrap_inline465





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:

eqnarray229


Since tex2html_wrap_inline467 :


Since tex2html_wrap_inline469 :


The product tex2html_wrap_inline471


Since tex2html_wrap_inline473 and tex2html_wrap_inline475


then we could say

eqnarray263


However the product of the first terms of the factors does not equal tex2html_wrap_inline477


Multiply tex2html_wrap_inline479 by tex2html_wrap_inline481


Let tex2html_wrap_inline483 s check to see if tex2html_wrap_inline485


eqnarray302


eqnarray315


eqnarray325


eqnarray331



The factors of tex2html_wrap_inline487 are tex2html_wrap_inline489 and tex2html_wrap_inline491






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