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Sunday, May 20, 2018

An equation involving radicals

While browsing the internet, the following brainteaser popped up on the screen: find x such that

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 b0 and b2a has as the only real solution:

x=(b2a2b)2

For the case a=b0, this reduces to


x=(a12)2

which is an integer if a is odd.

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