Question 77193
First part of the problem:
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Given:
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{{{x - 2*sqrt(x) = 0}}}
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Find all the values of x that satisfy this equation.
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The method involves getting the radical term on one side of the equation and the other term
on the other side.  We can do this by adding {{{2*sqrt(x)}}} to both sides and the equation
then becomes:
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{{{x = 2*sqrt(x)}}}
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Next square both sides. Note that when you do that, the right side squares as follows:
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{{{(2*sqrt(x))^2 = (2^2)*(sqrt(x))^2 = 4*x}}}
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and the equation therefore squares to:
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{{{x^2 = 4x}}}
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Subtract 4x from both sides and the equation is then:
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{{{x^2 - 4x = 0}}}
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Factor the common "x" from both terms on the left side to get:
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{{{x*(x - 4) = 0}}}
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Note that this equation will be true if either of the factors is zero.  Therefore,
one at a time we can set the factors equal to zero to solve the equation.  First:
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{{{x = 0}}}
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That is one answer. Then:
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{{{x-4 = 0}}}
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Solve by adding 4 to both sides to get:
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{{{x = 4}}}
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So the answer to the first part of this problem is that the equation given in the problem
will be satisfied by two values of x ... x = 0 and x = 4.
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For the second part of the problem the two graphs should look like this:
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{{{graph(600,400,-1,20,-5, 20, x, x - 2*sqrt(x))}}}
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Note that the original equation given in the problem does not permit x to be less than
zero because for real values you are not permitted to take the square root of a negative
number.
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The red line is the graph of {{{y = x}}} and the green line is the graph of {{{y = x - 2*sqrt(x)}}}
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Note that the green line is on the x-axis at the two values of x that we found previously,
x = 0 and x = +4.  Also note that the only place the two equations intersect is at the
origin. You can also show this is true mathematically by solving the equation set:
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{{{y = x}}} and {{{y = x - 2*sqrt(x)}}}
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Solve by substitution.  The first equation tells us that y equals x so we can substitute
x for y in the second equation.  With this substitution, the second equation becomes:
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{{{x = x - 2*sqrt(x)}}}
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Subtract x from both sides and the equation reduces to:
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{{{0 = -2*sqrt(x)}}}
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Square both sides and you get:
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{{{ 0 = 4*x}}}
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and divide both sides by 4 to find that:
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{{{x = 0}}}
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Then you can solve for the corresponding value of y by returning to the equation {{{y = x}}}
and plugging in 0 for x to find that y also = 0.  The only common point on the two graphs
is (0,0) ... the origin as is shown on the graphs themselves.  
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Hope this helps you to understand the problem and how to go about getting the solution 
to it.
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