Question 74094
{{{sqrt(2y)+7+4=y}}}
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Combine the 7 and the 4 on the left side to get:
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{{{sqrt(2y) + 11 = y}}}
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Next eliminate the 11 on the left side by subtracting 11 from both sides to get:
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{{{sqrt(2y) = y - 11}}}
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Now square both sides.  When you do you get:
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{{{2y = y^2 - 22y + 121}}}
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Next move everything to one side of the equation by subtracting 2y from both sides to get:
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{{{0 = y^2 - 24y + 121}}}
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Just to be in a little more conventional form, let's flip sides to get:
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{{{y^2 - 24y + 121 = 0}}}
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You can solve this by using the quadratic formula which says that for quadratic equations of
the standard form:
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{{{ax^2 + bx + c = 0}}}
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the solutions for x are:
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{{{x = (-b +- sqrt(b^2 - 4*a*c))/(2*a)}}}
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Now by comparing our equation with the standard form we can see that x = y, a = 1, b = -24, and
c= 121.
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Then just substitute these values into the equation for the solutions for x are:
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{{{y = (-(-24) +- sqrt((-24)^2 - 4*1*121))/(2*1)}}}
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All that remains to do is to simplify this:
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{{{y = (+24 +- sqrt(576 - 484))/2}}}
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The radical simplifies to {{{sqrt(92) = 9.591663}}}. Substituting that into the equation for
y results in:
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{{{y = (24 +- 9.591663)/2 }}}
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With the plus sign the numerator becomes 33.591663 and the value of y is:
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{{{y = 33.591663/2 = 16.7958315}}}
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and with the minus sign the numerator becomes: 24 - 9.591663 = 14.408337 and the value of y
is:
{{{y = 14.408337/2 = 7.2041685}}}
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Now you need to check these two answers to see if both of them work. First let's check to
see if y = 16.7958315 works in the original equation. Return to the problem and substitute
16.7958315 for y.  You get:
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{{{sqrt(2y)+7+4=y}}} which after the substitution for y becomes:
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{{{sqrt(2*16.7958315)+7+4=16.7958315}}}
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Subtract 11 from both sides results in:
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{{{sqrt(2*16.7958315) = 5.7958315}}}
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Work on the radical and you get:
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{{{ sqrt(33.59663) = 5.7958315}}} and taking the square root of the left side the equation becomes:
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5.7962600 = 5.7958315
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Close enough.  Differences are because of round offs.
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Now we need to check the other answer of y = 7.2041685.  Plug this value into the original equation to get:
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{{{sqrt(2*7.2041685)+7+4 = 7.2041685}}}
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Subtract -11 from both sides to get:
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{{{sqrt(14.408337) = -3.7958315}}}
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But wait a minute.  The square root of a positive number is not usually defined as being a
negative number. Only if your instructor says that both positive and negative values are 
to be considered will this solution work. In this case sqrt(14.408337) is allowed to
be -3.795831529 and the equation becomes -3.795831529  = -3.7958315 which also checks
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So both solutions work with the understanding that the square root of 2y is allowed to
be a negative number.
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Hope this helps you to understand the problem and why the problem requires that you check
both answers.  Only one of the answers that you got checks out.