Question

if x=3+2√2find x⁴+1/x⁴

Answer 1

x4−3x3−2x2−3x+1=0x4−3x3−2x2−3x+1=0

Divide by x2x2

We’ll get, (x2+1x2)−3(x+1x)=2(x2+1x2)−3(x+1x)=2

Now take y=(x+1x)y=(x+1x)

Therefore, (x2+1x2)=y2−2(x2+1x2)=y2−2

Substitute back !!

y2−2−3y=2y2−2−3y=2

y2−3y−4=0y2−3y−4=0

(y−4)(y+1)=0(y−4)(y+1)=0

y=−1,4y=−1,4

(x+1x)=−1,4(x+1x)=−1,4

x+1x=−1x+1x=−1

x2+x+1=0x2+x+1=0

x=−1±3√i2x=−1±3i2

x+1x=4x+1x=4

x2−4x+1=0x2−4x+1=0

x=4±12−−√2=2±3√x=4±122=2±3

All the solutions are,

x=2+3√,2−3√,−1+3√i2,−1+3√i2x=2+3,2−3,−1+3i2,−1+3i2

Where ‘ii’ is the imaginary unit (i=−1−−−√i=−1)




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