if a+b = 4 and a^2 + b^2 = 5, what is a^3 + b^3?

Answer 1

Let's start by cubing (a + b):
(a + b)^3 = a^3 + 3a²b + 3ab² + b^3

Solving for a^3 + b^3:
a^3 + b^3 = (a + b)^3 - (3a²b + 3ab²)

Factor out (a + b) from the right part of the equation:
a^3 + b^3 = (a + b)^3 - 3ab(a + b)

Next let's square (a + b):
(a + b)² = a² + 2ab + b²

Substituting in (a + b) = 4 and a² + b² = 5 we have:
4² = 5 + 2ab
16 = 5 + 2ab
11 = 2ab
ab = 11/2

So we know:
(a + b) = 4, ab = 11/2

Plugging it all back into the equation:
a^3 + b^3 = (a + b)^3 - 3ab(a + b)
a^3 + b^3 = 4^3 - 3(11/2)(4)
a^3 + b^3 = 64 - 66
a^3 + b^3 = -2

Answer 2

The hell with factoring. I'll use Brute Force!

a+b=4
b = 4-a
a^2 + b^2 = 5
a^2 + (4-a)^2 = 5
a^2 + 16 - 8a + a^2 = 5
2 a^2 - 8a + 16 = 5
2 a^2 - 8a + 11 = 0
a^2 - 4a + 5.5 = 0

a = 2 +/- sqrt(16-22)/2
a = 2 +/- i sqrt(3/2)
q = sqrt(3/2)
b = 4 - a

a1 = 2 + iq
b1 = 2 - iq

a2 = 2 - iq
b2 = 2 + iq

(a1)^2 = 4 + 4iq - q^2
(b1)^2 = 4 - 4iq - q^2

(a2)^2 = 4 - 4iq - q^2
(b2)^2 = 4 + 4iq - q^2

(a1)^3 = 8 + 8iq - 2 q^2 + 4iq - 4 q^2 - i q^3
(b1)^3 = 8 - 8iq - 2 q^2 - 4iq - 4 q^2 + i q^3

(a2)^3 = 8 - 8iq - 2 q^2 - 4iq - 4 q^2 + i q^3
(b2)^3 = 8 + 8iq - 2 q^2 + 4iq - 4 q^2 - i q^3

(a1)^3 + (b1)^3 = 16 - 12 q^2
(a2)^3 + (b2)^3 = 16 - 12 q^2

Case 1 and Case 2 give the same result.

a^3 + b^3 = 16 - 12(3/2) = 16 - 18 = -2

Answer 3

a^3 + b^3 = (a+b) x (a^2 + b^2 - ab) = (4) x (5 - ab) = 20 - 4ab.

But since a+b = 4,
we have (a+b)^2 = (a^2 + b^2) + 2ab = 4^2 = 16.

So 5 + 2ab = 16, or 2ab = 11, or 4ab = 22.

So a^3 + b^3 = 20 - 4ab = 20 - 22 = -2.

Answer 4

a^3+b^3 =(a+b)^3 -3ab(a+b) expressed in a+b as far aspossible
= 4^3 - 3ab* 4
= 64-12ab
= 64-6(2ab) because 2ab = (a+b)^2-(a^2+b^2) get in form
= 64-6((a+b)^2-(a^2+b^2))
= 64- 6*(4^2-5)
= 64 - 6*11
= -2

Answer 5

a + b = 4
Therefore,
(a + b)^2 = a^2 + 2ab + b^2 = 16
5 + 2ab = 16
2ab = 11
ab = 11/2

ab(a+b) = a^2b + ab^2
11/2(4) = a^2b + ab^2
a^2b + ab^2 = 22

a + b = 4
Therefore,
(a + b)^3=a^3 + 3a^2b + 3ab^2 + b^3 = 64
a^3 + b^3 + 3(a^2b + ab^2) = 64
a^3 + b^3 + 3(22) = 64
a^3 + b^3 = -2

Answer 6

It equals 6. Its a pattern 4, 5, 6

Answer 7

a^2 + b^2 = (a+b)^2 - 2ab => ab = [(a+b)^2 - (a^2+b^2)] / 2 = 5.5
a^3 + b^3 = (a+b) (a^2 + b^2 - ab) = 4 * (5-5.5) = -2