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$100^{-\lvert x \rvert} - x^2 = a^2$

I don't know how to approach this problem, due to the x in the exponent. I would appreciate hints more than outright solutions :)

John Doe
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1 Answers1

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I am assuming you are looking for solutions in real numbers. Think in terms of the intersection of two curves. $y=x^2+a^2$ (a parabola which lies on or above the $x-$axis) and $y=100^{-|x|}$ (even function). Now controlling $a$ gives you the ability to move your parabola up and down. Try to see what range of $a$ can help you maximize the number of intersections of the two curves.

See the picture:intersection of two curves

Anurag A
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  • I understand, so the answer would be from -1 to 1. I have tried to solve it right now using your advice but forgot the minus by the 100, so the answer was opposite. Wanted to find the mistake and your graph was handy. Thanks! – John Doe May 24 '15 at 21:55
  • @JohnDoe yes you are absolutely correct, $-1 < a < 1$ will guarantee two solutions. – Anurag A May 24 '15 at 21:56