if a + b = 7 & a*b=4 then (a^3 + b^3) = ?

Answer 1

a^3 + b^3 = (a+b)(a^2 + b^2 - ab)
a^2 + b^2 = (a+b)^2 - 2ab
So a^3 + b^3 = (a+b)[(a+b)^2 - 3ab]
Plugging in values we get
a^3 + b^3 = 7*(49-3*4) = 7(37) = 259

Answer 2

Recall that the sum of two cubes may be rewritten as:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Also: a^2 + b^2 = (a + b)^2 - 2ab

Substituting values:

a^2 + b^2 = 7^2 - 2*4 = 49 - 8 = 41

Therefore: a^3 + b^3 = 7*(41 - 4) = 7*37 = 259.

If you were to solve for a and b directly, you would then have to deal with two quadratic surds, plugged into the cubic expression a^3 + b^3 (which is certainly not an easy computational task).

Answer 3

a^3 + b^3 =
(a+b)(a^2 + b^2 - ab) but : a^2 + b^2 = (a+b)^2 - 2ab
therefore
a^3 + b^3 = (a+b)[(a+b)^2 - 2ab -1ab]
= (a+ b)[(a+b)^2 - 3ab]
a+b = 7 & a*b = 4
therefore subtituing the values:
a^3 + b^3 = 7 * [7^2 -12]
= 7*[49-12]
= 7 * 37
=259
ans : a^3 + b^3 =259

Answer 4

a^3 + b^3 =
... = (a + b)^3 - 3(a+b)ab
... = 7^3 - 3*7*4
... = 343 - 84
... = 259

Answer 5

b=4/a
a+4/a = 7
a^2-7a+4=0
a=(7+/-sqrt(49-16))/2
a=6.37228, b=0.62772
a=0.62772, b=6.37228
a^3+b^3=258.9998=259