SOLUTION: Solve each of the equations. Be sure to check your solution. {{{sqrt(2y)+7+4=y}}} Please help I am not sure how to solve this problem. I greatly appreciate it.

Algebra ->  Square-cubic-other-roots -> SOLUTION: Solve each of the equations. Be sure to check your solution. {{{sqrt(2y)+7+4=y}}} Please help I am not sure how to solve this problem. I greatly appreciate it.       Log On


   

Question 74094: Solve each of the equations. Be sure to check your solution.
sqrt%282y%29%2B7%2B4=y
Please help I am not sure how to solve this problem. I greatly appreciate it.
Found 2 solutions by checkley75, bucky:
Answer by checkley75(3666) About Me  (Show Source):
You can put this solution on YOUR website!
SQRT2Y+7+4=Y
SQRT2Y+11=Y
SQRT2Y=Y-11 NOW SQUARE BOTH SIDES OF THE EQUATION
2Y=Y^2-22Y+121
Y^2-22Y-2Y+121=0
Y^2-24Y+121=0
using the quadratic equation we have:
y=(24+-sqrt[576-4*1*121])/2*1
y=(24+-sqrt[576-484])/2
y=(24+-sqrt92)/2
y=(24+-9.59)/2
y=(24+9.59)/2
y=33.59/2
y=16.795 answer.
y=(24-9.59)/2
y=14.41/2
y=7.205 answer.

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%282y%29%2B7%2B4=y
.
Combine the 7 and the 4 on the left side to get:
.
sqrt%282y%29+%2B+11+=+y
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Next eliminate the 11 on the left side by subtracting 11 from both sides to get:
.
sqrt%282y%29+=+y+-+11
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Now square both sides. When you do you get:
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2y+=+y%5E2+-+22y+%2B+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%5E2+-+24y+%2B+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%5E2+-+24y+%2B+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:
.
ax%5E2+%2B+bx+%2B+c+=+0
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the solutions for x are:
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x+=+%28-b+%2B-+sqrt%28b%5E2+-+4%2Aa%2Ac%29%29%2F%282%2Aa%29
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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:
.
y+=+%28-%28-24%29+%2B-+sqrt%28%28-24%29%5E2+-+4%2A1%2A121%29%29%2F%282%2A1%29
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All that remains to do is to simplify this:
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y+=+%28%2B24+%2B-+sqrt%28576+-+484%29%29%2F2
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The radical simplifies to sqrt%2892%29+=+9.591663. Substituting that into the equation for
y results in:
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y+=+%2824+%2B-+9.591663%29%2F2+
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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%2F2+=+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%2F2+=+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:
.
sqrt%282y%29%2B7%2B4=y which after the substitution for y becomes:
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sqrt%282%2A16.7958315%29%2B7%2B4=16.7958315
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Subtract 11 from both sides results in:
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sqrt%282%2A16.7958315%29+=+5.7958315
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Work on the radical and you get:
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+sqrt%2833.59663%29+=+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%282%2A7.2041685%29%2B7%2B4+=+7.2041685
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Subtract -11 from both sides to get:
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sqrt%2814.408337%29+=+-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
.
So both solutions work with the understanding that the square root of 2y is allowed to
be a negative number.
.
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.