Question

Find all solutions y(real and otherwise)to the equation
y^8+16=17y^4

Answer 1

y=2, y=-2 y = 2i, y= -2i y = 1, y = -1 y = i, y = -i

Explanation:

y^8+16=17y^4

Subtract 17 y^4 from each side

y^8 - 17 y^4+16=17y^4-17 y^4

y^8 - 17 y^4+16 =0

Let x = y^4

x^2 -17x + 16 =0

Factor

( x-16) ( x-1) =0

Using the zero product property

x-16 =0 x-1 =0

x=16 x=1

Replace x with y^4

y^4 = 16 y^4 = 1

y^4 -16 = 0 y^4 -1 =0

Replace y^2 with z

z^2 -16 =0 z^2 -1 = 0

( z-4) ( z+4) = 0 ( z-1) = 0 ( z+1) =0

z=4 z=-4 z=1 z=-1

Replace z with y^2

y^2 = 4 y^2 = -4 y^2 = 1 y^2 = -1

Taking the square root

y=2, y=-2 y = 2i, y= -2i y = 1, y = -1 y = i, y = -i