Given x ^ 2 + x = 1, find the value of 2x ^ 4 + 4x ^ 3 + 2x ^ 2 + 8

Given x ^ 2 + x = 1, find the value of 2x ^ 4 + 4x ^ 3 + 2x ^ 2 + 8

Answer: the key is to split the 4x ^ 3 into 2x ^ 3 + 2x ^ 3x ^ 2 + x = 12x ^ 4 + 4x ^ 3 + 2x ^ 2 + 8 = 2x ^ 4 + 2x ^ 3 + 2 ^ X3 + 2x ^ 2 + 8 = 2 (x ^ 2 + x) x ^ 2 + 2 (x ^ 2 + x) x + 8 = 2x ^ 2 + 2x + 8 = 2 (x ^ 2 + x) + 8 = 2 + 2 + 8 = 10 or: x ^ 2 = 1-x, so: 2x ^ 4 + 4x ^ 3 + 2x ^ 3 + 2X ^ 2 + 8 = 2 = 2 (x ^ 2 + X + x ^ 2 + 8 = 2 (x ^ 2 + x ^ 2 + x ^ 2 + 8 = 2 (x ^ 2 + x ^ 2 + x ^ 2 + x ^ 2(x ^ 2 + 2x ^ 3 + 2x ^ 2 + 8 = 2 * x ^ 2 + 2 (x ^ 2 +...)

1. Given that 2x-8 and 4 are opposite to each other, find the value of 4x + 3

2x-8 and 4 are opposite numbers to each other
2x-8+4=0
2x=8-4
2x=4
X=2
4x+3=8+3=11

Given x = 1, find the value of 2x + 4 / x? 2 - 4x + 4 △ x? + 8 / 2x + 4 × (x? 2 - 4)

Replace x = 1 into
Original formula = 2 + 4-4 + 4 / 1 + 8 / 2 + 4 * (1-4)
=10-12
=-2

Given X / (x ^ 2-4x-1) = - 1 / 3, find the value of x ^ 4 + 2x + 1 / x ^ 5

x/(x^2-4x-1)=-1/3
x^2-4x-1=-3x
x^2-x-1=0
x^2=x+1
x^4=x^2+2x+1=x+1+2x+1=3x+2
x^5=x^4*x=(3x+2)x=3x^2+2x=3(x+1)+2x=5x+3
(x^4+2x+1)/x^5
=(3x+2+2x+1)/(5x+3)
=(5x+3)/(5x+3)
=1

Given that the square of x plus 4x minus one equals zero, find the value of the fourth power of 2x plus the third power of eight x minus the square of four minus eight x plus one worry

Equal to 0
Original formula = 2x (cubic of X + square of 4x-2x-4) + 1
=2x {x (square of X + 4x-1) - x-4} + 1
=2x(X*0-X-4)+1
=2X(-X-4)+1
=-The square of 2x - 8x + 1
=-2 (square of X + 4x-1)
=-2*0
=0

Given x ＾ 2 + 2x-1 = 0, find the value of X ＾ 3-2x + 4

Because x ＾ + 2x-1 = 0, the original formula = x (x ^ + 2x-1) - 2x ^ - x + 4 = - 2x ^ - 4x + 2 + 3x + 2 = - 2 (x ^ - 2x-1) + 3x + 2 = 3x + 2, from x ^ + 2x-1 = 0 to (x + 1) ^ = 2, so x = - 1 ± √ 2, so the original formula = 3 (- 1 ± √ 2) + 2 = - 1 ± 3 √ 2

Calculation question: ①(−1 3)−1+(−3)2×(π−2)0+(1 2)−3； ② Given x 2-5x-14 = 0, find the value of the algebraic expression - 2x (x + 3) + (2x + 1) 2 - (x + 1) (x + 2)

① The original formula = - 3 + 9 × 1 + 8 = 14;
② The original formula = - 2x2-6x + 4x2 + 4x + 1 - (x2 + 3x + 2) = x2-5x-1
∵x2-5x-14=0，
∴x2-5x=14，
The original formula = 14-1 = 13

Given the square of X + X-1 = 0, find the value of X's cubic power + 2x's square + 3

x²+x-1=0 ∴ x²+x=1
x³+2x²+3
=x³+x²+x²+3
=x（x²+x）+x²+3
=X + x 2 + 3 (brought in by x 2 + x = 1)
=1+3
=4

Given that: x? - X-1 = 0, then the value of - x? + 2x? + 2012 is_______ .

=-x³+x²+x+x²-x-1+2013
=-x(x²-x-1)+(x²-x-1)+2013
∵x²-x-1=0
The original formula = 0 + 0 + 2013
=2013

Given x 2 + x = 1, find the value of X quartic + x 3 - x 2 - 2x + 2005 It should be the value of X4 power + 2x 3 - x? - 2x + 2500

∵ x? + x = 1 ᙽ x? + X-1 = 0 according to the root formula x1.2 = (- 1 ± √ 5) / 2 ⁴ x ⁴ x ⁴ x ⁴ x ⁴ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x ∵ x = 1 9 X-1 9 X-1