Question 72601
First Problem:
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Solve the equation {{{x - sqrt(x) = 0}}} for all values of x that will satisfy the equation:
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First add {{{sqrt(x)}}} to both sides of the equation.  When you do, the equation becomes:
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{{{x = sqrt(x)}}}
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Next square both sides:
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{{{x^2 = (sqrt(x))^2}}} which becomes {{{x^2 = x}}}
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Then subtract x from both sides to get:
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{{{x^2 - x = 0}}}
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On the left side factor an x:
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{{{x*(x-1) = 0}}}
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Note that this equation will be true if either of the factors on the left side equals zero.
This is true because if one of the factors on the left side is zero, the entire left
side will be zero and therefore will equal the right side. So set each of the two factors
equal to zero and you get:
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{{{x = 0}}} and
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{{{x-1 =0}}} and by adding 1 to both sides of this equation you get {{{x = 1}}}.
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So there are two answers to this equation: {{{x = 0}}} and {{{x = 1}}}.  If you substitute
0 for x and if you next substitute 1 for x in the original equation you will see that these
two answers work. 
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Next graph y = x and y = sqrt(x) on the same set of axes:

{{{graph(800, 800,-3,6,-3,6,x, sqrt(x))}}}
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The green graph represents the y = square root of x and the red graph represents y = x. Note
that the green graph should hit the origin, but there is a flaw in the drafting program that
does not do that.  So you will have to make your version of the green graph go down to the
origin.
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What is the connection where the graphs cross.  The crossing points occur at x = 0 and x = 1
which are the answers we got for the problem.  At those two points x and the square root of
x are equal so that when you subtract them, the answer is zero.  
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Hope this helps you to understand the problem.