Question

Solve for all complex roots (2x^10)+(3x^9)-(5x^8)-(9x^7)-(x^6)+(3x^5)+(7x^4)+(9x^3)-(x^2)-(6x)-2

Answer 1

complex roots involve square roots of negative numbers

when writing complex roots, we use i where i=√-1

in your question, technically, we can't solve for x since it isn't equal to anything

if we assume that it's equal to 0, then

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

if we factor it using a factoring aid (because that's hard to factor) we get

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

set each to 0

(x-1)²=0

x-1=0

x=1, not comlex

(x+1)³=0

x+1=0

x=-1, not a comlex root

(2x+1)=0

2x+1=0

2x=-1

x=-1/2, not a complex root

x²-2=0

x²=2

$x=\pm \sqrt{2}$, not a complex root

x²+1=0

x²=-1

$x=\pm \sqrt{-1}$

$x=\pm i$ are complex roots

the 2 complex roots are i and -i where i=√-1

Answer 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:

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