√(x) + √(x-36) = 2 Find all Real and imaginary solutions?
Can someone explain step by step how to solve this? I've tried all the methods in the book and none of them seem to work. :/ Wolfgram alpha doesn't even explain it well enough.
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Square the equation:
(√(x))² + 2√(x)√(x-36) + (√(x-36))² = 4;
x + 2√(x)√(x-36) + x - 36 = 4;
2√(x)√(x-36) = 40 - 2x.
Square the equation again:
4x(x-36) = 1600 - 160x + 4x²;
4x² - 144x = 1600 - 160x + 4x²;
the 4x² cancels:
-144x = 1600 - 160x
16x = 1600;
x = 100.
When doing equations with radicals, we must check the answer:
√(100) + √(100-36) = √(100) + √(64) = 10 + 8 = 18.
Since we didn't get 2 as the equation says, then we must conclude that the equation has no solution. In fact, since the value of x in the second term can't be less than 36, then the value on the right side of the equation has to be 6 or greater in order for the equation to have a solution.