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sqrt(x+3) = x+1
\n" ); document.write( "square both sides to get:
\n" ); document.write( "x+3 = (x+1)^2
\n" ); document.write( "subtract (x+3) from both sides of the equation to get:
\n" ); document.write( "(x+1)^2 - (x+3) = 0
\n" ); document.write( "expand the equation to get:
\n" ); document.write( "x^2 + 2x + 1 - (x+3) = 0
\n" ); document.write( "simplify to get:
\n" ); document.write( "x^2 + 2x + 1 - x = 3 = 0
\n" ); document.write( "combine like terms to get:
\n" ); document.write( "x^2 + x - 2 = 0
\n" ); document.write( "factor to get:
\n" ); document.write( "(x + 2) * (x - 1) = 0
\n" ); document.write( "solve for x to get:
\n" ); document.write( "x = -2
\n" ); document.write( "x = 1
\n" ); document.write( "you have 2 possible values of x that will satisfy the equation of x^2 + x - 2 = 0
\n" ); document.write( "if you go back to your original equation, you will get:
\n" ); document.write( "sqrt(x+3) = x+1
\n" ); document.write( "when x = -2, this equation becomes sqrt(1) = -1
\n" ); document.write( "when x = 1, this equation becomes sqrt(4) = 2
\n" ); document.write( "sqrt(4) = 2 is a valid answer.
\n" ); document.write( "sqrt(1) = -1 is not because the square root of 1 is equal to 1, not -1.
\n" ); document.write( "the equation of sqrt(x+3) = x+1 only looks for the primary root which is the positive root.
\n" ); document.write( "while the negative root is a valid answer if the equation was posed another way, it is not valid the way the question was posed.
\n" ); document.write( "when you see an original equation like sqrt(x) = y, then y has to be positive.
\n" ); document.write( "when you see something like x^2 = y^2, then sqrt(x) = +/- y is valid, meaning that y can be positive or negative.
\n" ); document.write( "if your question was original posed as:
\n" ); document.write( "x+3 = (x+1)^2, then you would still have gotten x = 1, x = -2, only this time x = -2 would have been valid.
\n" ); document.write( "now when you plug those values into the original equation, you get:
\n" ); document.write( "x+3 = (x+1)^2 becomes:
\n" ); document.write( "4 = 4 when x = 1
\n" ); document.write( "1 = 1 when x = -2
\n" ); document.write( "there is no conflict where you got sqrt(x+3) = x+1 becomes 1 = -1 because you could only take the positive square root (the principal square root).
\n" ); document.write( "here's another reference on principal square root that addresses that problem.
\n" ); document.write( "http://www.mathpath.org/concepts/principal.square.root.htm
\n" ); document.write( "bottom line is that there is a convention in mathematics as follows:
\n" ); document.write( "if the equation says something like x^2 = something, then plus and minus roots are allowed.
\n" ); document.write( "if the equation says something like sqrt(x) = something, then only the positive roots are allowed, even though the negative roots are just as valid.
\n" ); document.write( "the reference mentions an inverse function.
\n" ); document.write( "an example is y = x^2
\n" ); document.write( "the graph of that equation would be:
\n" ); document.write( "
\n" ); document.write( "to derive the inverse function, we solve for x and then replace x with y and replace y with x.
\n" ); document.write( "our equation of y = x^2 becomes:
\n" ); document.write( "x^2 = y
\n" ); document.write( "x = +/- sqrt(y)
\n" ); document.write( "we replace x with y and y with x to get:
\n" ); document.write( "y = +/- sqrt(x)
\n" ); document.write( "a graph of this equation gets us:
\n" ); document.write( "
\n" ); document.write( "while this is accurate, it is not a function because you can have multiple values of y for the same value of x (2 values of y for each value of x to be exact).
\n" ); document.write( "so we only take the positive square root of y to get:
\n" ); document.write( "y = sqrt(x).
\n" ); document.write( "our domain had to be positive to start with because sqrt(-x) is invalid, i.e. you can't take the square root of a negative number.
\n" ); document.write( "so, only values of x >= 0 are valid for the inverse equation.
\n" ); document.write( "suppose, then, if x = 4.
\n" ); document.write( "y = sqrt(x) = sqrt(4) = 2
\n" ); document.write( "y does not equal -2, even though (-2)^2 = 4.
\n" ); document.write( "we only took the positive square root of y, or the principal root of y, and we did not allow the negative square root of y.
\n" ); document.write( "y = -2 is valid, but, by convention, we don't allow it, because then our inverse function is not really a function, but a relation.
\n" ); document.write( "this makes the root of y = -2 invalid.
\n" ); document.write( "the inverse equation now becomes y = sqrt(x), rather than y = +/- sqrt(x).
\n" ); document.write( "a graph of that equation is shown below:
\n" ); document.write( "
\n" ); document.write( "note that the inverse equation is now a function because there is only 1 value of y for each value of x.
\n" ); document.write( "FOR PRACTICAL PURPOSES, YOU JUST HAVE TO REMEMBER:
\n" ); document.write( "y^2 = x leads to a solution of y = +/- sqrt(x).
\n" ); document.write( "y = sqrt(x) leads to a solution of y = + sqrt(x) only.
\n" ); document.write( "your equation started off as:
\n" ); document.write( "sqrt(x+3) = x+1
\n" ); document.write( "this means that you are to get the principal (positive) root only.
\n" ); document.write( "that is why the root of x = -2 was invalid, and therefore extraneous.
\n" ); document.write( "here's a couple of references that address the extraneous root issue:
\n" ); document.write( "http://www.mathwords.com/e/extraneous_solution.htm
\n" ); document.write( "http://mathcentral.uregina.ca/qq/database/qq.09.02/paul2.html
\n" ); document.write( "http://mathmistakes.info/facts/AlgebraFacts/learn/ctm/extra.html
\n" ); document.write( "these cases point up invalid solutions.
\n" ); document.write( "sometimes the solutions are valid, but can't be used because of domain restrictions.
\n" ); document.write( "the case of the length of a side would be such a restriction.
\n" ); document.write( "if the value of x has to be positive, then clearly a negative value of x would be invalid.
\n" ); document.write( "also the domain might now allow a division by 0, so if the value of x obtained as a solution makes the original equation be divided by 0, then that solution is also invalid.
\n" ); document.write( "always check your solutions against the ORIGINAL equation.
\n" ); document.write( "the particular equation you posed does not appear to have an extraneous solution unless there are some restrictions you didn't provide.\r
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