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Algebra Level 3

Suppose that m 2 − 10 m + 1 = 0 m^2-\sqrt{10}m+1=0 . Determine m 4 + m − 4 m^4+m^{-4} .


The answer is 62.

Tom Zhou by Tom Zhou , 21, Canada

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1 solution

Justin Dong
Aug 31, 2014

Note that m + 1 m = m 2 + 1 m m+\frac{1}{m} = \frac{m^2+1}{m}

Adding 10 m \sqrt{10}m to both sides of the given equation, we have m 2 + 1 = 10 m . m^2+1=\sqrt{10}m.

From above, m 2 + 1 m = 10 m m = 10 . \frac{m^2+1}{m}=\frac{\sqrt{10}m}{m}=\sqrt{10}.

Then ( m + 1 m ) 2 − 2 = m 2 + 1 m 2 = 8 (m+\frac{1}{m})^2-2=m^2+\frac{1}{m^2}=8

( m 2 + 1 m 2 ) 2 − 2 = m 4 + 1 m 4 = 62 . (m^2+\frac{1}{m^2})^2-2=m^4+\frac{1}{m^4}=\boxed{62}.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

wow!! really short way and avoiding using quadratic formula (y)

Lakshay Sethi - 6 years, 9 months ago

NOT THOUROUGHLY EXPLAINED ENOUGH

Eug Torres - 6 years, 8 months ago

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