SOLUTION: Solve symbolically: Square root of (5-x) + square root of (4x) = 5 I know you have to square both sides but I don't know how to multiply the different radicals.

Algebra ->  Square-cubic-other-roots -> SOLUTION: Solve symbolically: Square root of (5-x) + square root of (4x) = 5 I know you have to square both sides but I don't know how to multiply the different radicals.      Log On


   

Question 545455: Solve symbolically:
Square root of (5-x) + square root of (4x) = 5
I know you have to square both sides but I don't know how to multiply the different radicals.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
You are given to solve for x:
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sqrt%285-x%29+%2B+sqrt%284x%29=5
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It always helps me to put the radicals on different sides of the equation. In this case let's subtract sqrt%284x%29 from both sides to make the equation become:
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sqrt%285+-+x%29+=+5+-+sqrt%284x%29
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Now square both sides:
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%28sqrt%285+-+x%29%29%5E2+=+%285-sqrt%284x%29%29%5E2
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When you square a square root, the answer is just the term under the radical sign. So the left side changes as shown below:
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5+-+x+=+%285-sqrt%284x%29%29%5E2
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We can square the right side by doing a FOIL multiplication as shown:
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5+-+x+=+%285-sqrt%284x%29%29%2A%285-sqrt%284x%29%29
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The FOIL multiplication results in:
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5+-+x+=+25+-5%2Asqrt%284x%29+-5%2Asqrt%284x%29+%2B+4x
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On the right side combine the two radical terms. Then group the two non-radical terms. This will give you:
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5+-+x+=+25+%2B4x+-10%2Asqrt%284x%29
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Get all the non-radical terms on one side of the equation. You can do this by subtracting 25 and 4x from both sides. This leaves the radical by itself on the right side and the left side becomes:
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5+-+x+-+25+-4x+=+-10%2Asqrt%284x%29
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Combine the like terms on the left side (5 and -25 = -20, and -x and -4x = -5x) to get:
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-20+-+5x+=+-10%2Asqrt%284x%29
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Since all the terms are negative let's multiply both sides (all terms) by -1 to change the equation to:
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20+%2B+5x+=+10%2Asqrt%284x%29
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Now square both sides again. This involves another FOIL multiplication on the left side and squaring a radical (and the 10) on the right side so that you have:
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%2820+%2B+5x%29%2A%2820%2B5x%29+=+100%2A4x
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The FOIL multiplication results in:
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400+%2B100x+%2B100x+%2B+25x%5E2+=+100%2A4x
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Combine the two x terms on the left side and multiply out the right side to get:
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400+%2B+200x+%2B+25x%5E2+=+400x
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Subtract 400x from both sides to get rid of the 400x on the right side:
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400+-200x+%2B+25x%5E2+=+0
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Simplify by dividing all terms on both sides by 25 to get:
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16+-+8x+%2Bx%5E2+=0
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Rearrange the terms on the left side in descending order of the power of x:
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x%5E2+-8x+%2B+16+=+0
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The left side factors:
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%28x+-4%29%5E2+=+0
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To make the left side equal to the right side it will require that x-4 = 0. Solving this (by adding 4 to both sides) tells you that x = +4 is the solution to this problem.
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You can check this solution by substituting +4 for x in the original problem:
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sqrt%285-x%29+%2B+sqrt%284x%29=5
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becomes:
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sqrt%285-%28%2B4%29%29+%2B+sqrt%284%2A4%29=5
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which becomes:
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sqrt%281%29+%2B+sqrt%2816%29=5
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and this simplifies to:
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1+%2B+4+=+5
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It checks. And we can say with confidence that x = 4 is the solution to this problem.
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Hope this helps you to understand the problem and helps you to see how to multiply out the radicals.
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