Question

已知a、b满足a+b=1，a²+b²=2，求a³+b³，a的4次方+b的4次方的值

Answer 1

a+b=1，a²+b²=2

(a+b)^2=1
a^2+b^2+2ab=1
ab=-1/2

a³+b³

=(a+b)(a^2+b^2-ab)
=(a+b)[(a+b)^2-3ab)
=-1/2
a的4次方+b的4次方

=(a^2+b^2)-2a^2b^2
=1-1/2
=1/2

Answer 2

a+b=1

(a+b)²=1
a²+2ab+b²=1

2+2ab=1
ab = -1/2

a³+b³

=(a+b)(a²-ab+b²)
=1×(2-1/2)
=3/2

a²+b²=2

(a²+b²)²=4

a的4次方 + 2a²b² + b的4次方 = 16
a的4次方+b的4次方 + 2(3/2)² = 16

a的4次方+b的4次方 = 7/2

Answer 3

a+b=1
(a+b)^2=1
a^2+b^2+2ab=1
2+2ab=1
2ab=-1
ab=-1/2

a^3+b^3
=(a+b)(a^2-ab+b^2)
=1*[2-(-1/2)]
=5/2

a^4+b^4
=a^4+b^4+2a^2b^2-2a^2b^2
=(a^2+b^2)^2-2(ab)^2
=2^2-2*(-1/2)^2
=4-1/2
=7/2