SOLUTION: Can someone show me how this is solved algebraically? "If you take half of my earnings, and then subtract the square root of my earnings, the result is 24." Step 1. Set up eq

Algebra ->  Square-cubic-other-roots -> SOLUTION: Can someone show me how this is solved algebraically? "If you take half of my earnings, and then subtract the square root of my earnings, the result is 24." Step 1. Set up eq      Log On


   

Question 985456: Can someone show me how this is solved algebraically?
"If you take half of my earnings, and then subtract the square root of my earnings, the result is 24."
Step 1. Set up equation: x/2 - the square root of x = 24
But with the fraction and the square root, I'm not sure how to do the math correctly. I know I need to remove the fraction and then square the equation to get rid of the radical.
I can get the answer by plugging from the answer choices, but how do I get it algebraically?
Thank you.
Helen

Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
x, your earnings.

x%2F2, half of your earnings
sqrt%28x%29, square root of your earnings

x%2F2-sqrt%28x%29, take half of your earnings and subtract the square root of your earnings

".... is 24."
highlight%28x%2F2-sqrt%28x%29=24%29
Solve.

Best if you first isolate the irrational expression on one side of the equation first.
TRY. Do the work.
.
.
x%5E2-100x%2B2304=0
-
Either x=36 or x=64.
One of these will work and the other will fail.

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Another tutor fully solved this for you.

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!

x%2F2 - sqrt%28x%29 = 24,

x%2F2+-+24 = sqrt%28x%29,

%28x%2F2+-+24%29%5E2 = %28sqrt%28x%29%29%5E2

%28x%2F2%29%5E2+-+2%2A%28x%2F2%29%2A24+%2B+24%5E2 = x,

x%5E2%2F4+-+24x+-+x+%2B+24%5E2 = 0, ---> (multiply both sides by 4 ) --->

x%5E2+-+100x+%2B+4%2A24%5E2 = 0,

x%5E2+-+100x+%2B+2304 = 0.

x%5B1%5D = %28100%2Bsqrt%28100%5E2-4%2A2304%29%29%2F2 = %28100%2Bsqrt%28784%29%29%2F2 = %28100%2B28%29%2F2 = 128%2F2 = 64,

x%5B2%5D = %28100-sqrt%28100%5E2-4%2A2304%29%29%2F2 = %28100-sqrt%28784%29%29%2F2 = %28100-28%29%2F2 = 72%2F2 = 36.

Check. 1.  64%2F2+-+sqrt%2864%29 = 32+-+8 = 24.     Yes!

          2.  36%2F2+-+sqrt%2836%29 = 18+-+6 = 12.      No!

Only the root 64 satisfies the given condition.

Answer.  My earning is  64.   (64 of what ? :)