# Try to solve root in inequality got wrong result

I am very confused. So I have to solve this inequality. The result is $13/24$.

But if I try to solve it myself, I get $17/24$. Because:

$$\sqrt{\left(\frac{-5}{24}\right)^2 + \frac{1}{4}} = \frac{5}{24} + \frac{1}{2} = \frac{5}{24} + \frac{12}{24} = \frac{17}{24}.$$

The right solution should be $13/24$. Where is my failure?

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$\sqrt{a^2+b^2}\ne a+b$ –  lab bhattacharjee Oct 25 '12 at 14:40
You’re not solving an inequality: you’re simplifying the expression with the square root. –  Brian M. Scott Oct 25 '12 at 14:45
Oh sorry, that's right. It was a part of my inequality –  bob morane Oct 25 '12 at 14:47

$\sqrt{(-5/24)^2+1/4} = \sqrt{25/24^2+1/4} = \sqrt{(25+6\cdot 24)/24^2} = \sqrt{169}/\sqrt{24^2} = 13/24$
@bobmorane, that's right. the trick is to get it into the form $\sqrt{a^2}/\sqrt{b^2} = a/b$ –  sperners lemma Oct 25 '12 at 14:53