SOLUTION: 2 times the square root of x minus 1 minus the square root of 3x minus 5 equals the square root of x minus 9

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Question 202564: 2 times the square root of x minus 1 minus the square root of 3x minus 5 equals the square root of x minus 9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
2%2Asqrt%28x-1%29-sqrt%283x-5%29=sqrt%28x-9%29 Start with the given equation.


%282%2Asqrt%28x-1%29-sqrt%283x-5%29%29%5E2=%28sqrt%28x-9%29%29%5E2 Square both sides


%282%2Asqrt%28x-1%29-sqrt%283x-5%29%29%5E2=x-9 Square the square root.


4%28x-1%29-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29%2B3x-5=x-9 FOIL


4x-4-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29%2B3x-5=x-9 Distribute


7x-9-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=x-9 Combine like terms.


-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=x-9%2B9-7x Get all the non-radical terms to one side


-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=-6x Combine like terms.


-4%2Asqrt%28%28x-1%29%283x-5%29%29=-6x Combine the roots.


sqrt%28%28x-1%29%283x-5%29%29=%28-6x%29%2F%28-4%29 Divide both sides by -4.


sqrt%28%28x-1%29%283x-5%29%29=%283%2F2%29x Reduce


%28sqrt%28%28x-1%29%283x-5%29%29%29%5E2=%28%283%2F2%29x%29%5E2 Square both sides


%28x-1%29%283x-5%29=%28%283%2F2%29x%29%5E2 Square the square root


%28x-1%29%283x-5%29=%289%2F4%29x%5E2 Square the right side


4%28x-1%29%283x-5%29=9x%5E2 Multiply both sides by 4.


4%283x%5E2-8x%2B5%29=9x%5E2 FOIL


12x%5E2-32x%2B20=9x%5E2 Distribute


12x%5E2-32x%2B20-9x%5E2=0 Subtract {{9x^2}}} from both sides.


3x%5E2-32x%2B20=0 Combine like terms.


Notice that the quadratic 3x%5E2-32x%2B20 is in the form of Ax%5E2%2BBx%2BC where A=3, B=-32, and C=20


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%28-32%29+%2B-+sqrt%28+%28-32%29%5E2-4%283%29%2820%29+%29%29%2F%282%283%29%29 Plug in A=3, B=-32, and C=20


x+=+%2832+%2B-+sqrt%28+%28-32%29%5E2-4%283%29%2820%29+%29%29%2F%282%283%29%29 Negate -32 to get 32.


x+=+%2832+%2B-+sqrt%28+1024-4%283%29%2820%29+%29%29%2F%282%283%29%29 Square -32 to get 1024.


x+=+%2832+%2B-+sqrt%28+1024-240+%29%29%2F%282%283%29%29 Multiply 4%283%29%2820%29 to get 240


x+=+%2832+%2B-+sqrt%28+784+%29%29%2F%282%283%29%29 Subtract 240 from 1024 to get 784


x+=+%2832+%2B-+sqrt%28+784+%29%29%2F%286%29 Multiply 2 and 3 to get 6.


x+=+%2832+%2B-+28%29%2F%286%29 Take the square root of 784 to get 28.


x+=+%2832+%2B+28%29%2F%286%29 or x+=+%2832+-+28%29%2F%286%29 Break up the expression.


x+=+%2860%29%2F%286%29 or x+=++%284%29%2F%286%29 Combine like terms.


x+=+10 or x+=+2%2F3 Simplify.


So the possible solutions are x+=+10 or x+=+2%2F3

However, if you plug in x+=+2%2F3 into the original equation, you'll get a contradiction.

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Answer:


So the only solution is x+=+10