Calculate Equations

Algebra Level 2

Positive real numbers a , b a,b are such that x = 2 a b b 2 + 1 x=\frac{2ab}{b^2+1} .

Is the value of Q = a + x + a − x a + x − a − x + 1 3 b Q=\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}} + \frac{1}{3b} constant?

Note: Assume a , b a,b are given numbers, therefore constant.

No Paradoxical Answer. Yes Sometimes.

Tin Le by Tin Le , 15, Vietnam

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1 solution

a + x = ( b + 1 ) a b 2 + 1 \sqrt {a+x}=(b+1)\sqrt \dfrac{a}{b^2+1} and a − x = ( b − 1 ) a b 2 + 1 \sqrt {a-x}=(b-1)\sqrt \dfrac{a}{b^2+1} . So the given quantity is b + 1 3 b = c o n s t a n t b+\dfrac{1}{3b}=\boxed {constant} provided b b is constant

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