Solve for all complex roots (2x^10)+()+(3x^5)+(7x^4)+()-2

Solve for x by completing the square:

2 x^10 + 3 x^9 - 5 x^8 - 9 x^7 - x^6 + 3 x^5 + 7 x^4 + 9 x^3 - x^2 - 6 x - 2 = 0

The left hand side factors into a product with five terms:

(x - 1)^2 (x + 1)^3 (2 x + 1) (x^2 - 2) (x^2 + 1) = 0

Split into five equations:

(x - 1)^2 = 0 or (x + 1)^3 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0

Take the square root of both sides:

x - 1 = 0 or (x + 1)^3 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0

Add 1 to both sides:

x = 1 or (x + 1)^3 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0

Taking cube roots gives 0 times the third roots of unity:

x = 1 or x + 1 = 0 or x + 1 = 0 or x + 1 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0

Subtract 1 from both sides:

x = 1 or x = -1 or x + 1 = 0 or x + 1 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0

Subtract 1 from both sides:

x = 1 or x = -1 or x = -1 or x + 1 = 0 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0

Subtract 1 from both sides:

x = 1 or x = -1 or x = -1 or x = -1 or 2 x + 1 = 0 or x^2 - 2 = 0 or x^2 + 1 = 0

Subtract 1 from both sides:

x = 1 or x = -1 or x = -1 or x = -1 or 2 x = -1 or x^2 - 2 = 0 or x^2 + 1 = 0

Divide both sides by 2:

x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x^2 - 2 = 0 or x^2 + 1 = 0

Add 2 to both sides:

x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x^2 = 2 or x^2 + 1 = 0

Take the square root of both sides:

x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x = sqrt(2) or x = -sqrt(2) or x^2 + 1 = 0

Subtract 1 from both sides:

x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x = sqrt(2) or x = -sqrt(2) or x^2 = -1

Take the square root of both sides:

x = 1 or x = -1 or x = -1 or x = -1 or x = -1/2 or x = sqrt(2) or x = -sqrt(2) or x = i or x = -i

There are {2} duplicate solutions:

x = -1 or x = 1 or x = -1/2 or x = -i or x = i or x = sqrt(2) or x = -sqrt(2)

hello from mrbilldoesmath!

answer:

25x^2 + 90x + 84

discussion:

mn(x) = m(n(x)) = m( 5x + 9) =

(5x + 9)^2 + 3 =>

25x^2 + 90x + 9^2 + 3 =

25x^2 + 90x + 84                 (as 9^2 = 9*9 = 81)

which is the second choice

regards,  

mrb

p.s.   i'll be on vacation from friday, dec 22 to jan 2, 2019. have a great new year!