√x+15+√x=15
A straight forward approach is to square both sides, move the single term with radicals to one side and square again, and reducing terms to obtain a linear equation in x whose solution gives the answer.
Here is another (somewhat simpler) way to solve this problem. First note that the left hand side is a increasing function of x and at x=0 is equal to √15<15, so there is a single real solution to the equation above.
If we set x=y2 and x+15=(y+z)2, we get 15=2yz+z2=z(2y+z). The left hand side of the original equation then becomes:
y+z+y=2y+z=15. Combine this with the above it follows that z=1 and y=7. Thus the answer is x=49.
In general, this method shows that the equation
√x+a+√x=b where b≥0 and b2≥a has as the only real solution:
x=(b2−a2b)2
For the case a=b≥0, this reduces to
x=(a−12)2
which is an integer if a is odd.
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