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Can you help me to do this question:it is from a past cambridge exam paper

Find the positive constants $a$ and $b$ such that $x^4+9/x^4 =[x^2-a/x^2 ]^2+b$ for all non-zero values of $x$.

Hence write down, or obtain otherwise, the least possible value of $x^4+9/x^4$ for real values of $x$.

I tried to expand $[x^2-a/x^2 ]^2$ and compare it with $x^4+9/x^4$ and I got $a=3$ but I don't know how to find $b$ and how to find the least possible value of $x^4+9/x^4$.

Pedro
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Sophia Granger
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1 Answers1

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So I assume you've done $LHS =\left( {x^2- \frac a {x^2}} \right)^2 = x^4 - \left({x^2} \right) \left(\frac a {x^2} \right) - \frac {a^2} {x^4}$

The middle term simplifies to $a$ and you've said you already have $a=3$

You should be able to do the rest...

tomi
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