if a + b = 5 & a*b=3 then (a^3 - b^3) = ?

Answer 1

now ( a - b)^2 = (a +b )^2-4ab =5^2 - 4*3 = 13
then a-b = sqrt(13)
again , a^3 - b^3 = (a-b)^3 + 3ab(a-b) = 13^(3/2) + 3*3*sqrt(13)
= sqrt(13) [ 13 + 9]
= 22*sqrt (13)

Answer 2

If a-b = 5, then only it is possible to solve this as follows:

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

a^3 - b^3 = (a-b)^3 + 3*a^2*b - 3*a*b^2
= (a-b)^3 + 3*a*b*(a-b)
= 5^3 + 3*3*5
= 125 + 45
= 170

Answer 3

there are two possible answers, but they're just generally the same answers, but with different signs.

if you use the quadratic formula on the system of equations above, you would get two roots (values) for a: (5 + sqrt(5)) / 2 and (5 - sqrt(5)) / 2. with these values for a, you would see that the roots for b then are the same as a, except that they're the opposite signs (if a = (5 + sqrt(5)) / 2 then b = (5 - sqrt(5)) / 2 )

if you plug these values into the third equation then, you would see (after a lot of cancelling out of values) that the third equation also yields two values, depending on what values of a and b you use.

the values of the third equation therefore, are ( 20 * sqrt(5) ) and ( -20 * sqrt(5) ) ^_^

Answer 4

a + b = 5
ab = 3

a + b = 5
b = -a + 5

a(-a + 5) = 3
-a^2 + 5a = 3
-a^2 + 5a - 3 = 0
-(a^2 - 5a + 3) = 0
a^2 - 5a + 3 = 0

a = (5 ± sqrt(25 - 12))/2
a = (5 ± sqrt(13))/2

b = -(5 ± sqrt(13))/2 + 5
b = (-5/2) ± (-1/2)sqrt(13) + 5
b = (-5/2) + (10/2) ± (-1/2)sqrt(13)
b = (5/2) ± (-1/2)sqrt(13)
b = (-1/2)(-5 ± sqrt(13))

a = (1/2)(5 + sqrt(13))
b = (1/2)(5 - sqrt(13))

((1/2)(5 + sqrt(13)))^3 - ((1/2)(5 - sqrt(13)))^3

(1/8)((5 + sqrt(13))(5 + sqrt(13))(5 + sqrt(13)) - (1/8)((5 - sqrt(13))(5 - sqrt(13))(5 - sqrt(13)))

(1/8)((25 + 5sqrt(13) + 5sqrt(13) + 13)(5 + sqrt(13))) - (1/8)((25 - 5sqrt(13) - 5sqrt(13) + 13)(5 - sqrt(13)))

(1/8)((38 + 10sqrt(13))(5 + sqrt(13))) - (1/8)((38 - 10sqrt(13))(5 - sqrt(13)))

(1/8)(190 + 38sqrt(13) + 50sqrt(13) + 10(13)) - (1/8)(190 - 38sqrt(13) - 50sqrt(13) + 10(13))

(1/8)(190 + 88sqrt(13) + 130) - (1/8)(190 - 88sqrt(13) + 130)

(1/8)(320 + 88sqrt(13)) - (1/8)(320 - 88sqrt(13))

40 + 11sqrt(13) - 40 + 11sqrt(13)

22sqrt(13)

ANS :

a^3 - b^3 = 22sqrt(13)

Answer 5

a^3 - b^3 = (a-b)(a^2+ab+b^2)
a^2+ab+b^2 = (a+b)^2 - ab
substituting the second equation in the first
a^3 - b^3 = (a-b)((a+b)^2 - ab)
= (a-b)(25 - 3) = 22(a-b)---------(1)

(a-b)^2 = (a+b)^2 - 4ab = 25 - 12 = 13
(a-b) = sqrt 13 substituing this value in equation (1)
a^3 - b^3 = 22(sqrt13)

Answer 6

(a - b)^2 = (a + b)^2 - 4ab = 25 - 12 = 13,
so a - b = sqrt 13.

a^3 - b^3 =
... = (a - b) * (a^2 + ab + b^2)
... = (a - b) * ([a + b]^2 - ab)
... = sqrt 13 * (25 - 3) = 22 sqrt 13

Fun problem!

Answer 7

a+b=5---------(1)
a*b=3----------(2)
from (1) b = 5-a
substituting above eqn in (2)
a(5-a)=3
5a-a.a=3
a.a-5a+3=0
therfore a = 5(+/-)sqrt(13)/2
from (1) a = 5-b
substituing above eqn in (2)
(5-b)b=3
5b-b.b=3
b.b-5b+3=0
therfore b = 5(+/-)sqrt(13)/2
Hence Ans = 0 (if u consider both as positive or both as negative. If u consider one as +ve and other as -ve then do the simplification by urself)

Answer 8

22 sqrt 13

Answer 9

a+b = 5
ab= 3

a= 3/b

3/b + b = 5
(3+b^2)/b=5
3+b^2=5b
b^2-5b+3=0

b = ((5+ (13^.5))/2) or ((5- (13^.5))/2)
a = 6/(5+13^.5) or 6/(5-13^.5)

substituting

a= 6/(5+13^.5) , b=((5+ (13^.5))/2)
(a^3 - b^3) = 79.32212806

a= 6/(5-13^.5), b= ((5- (13^.5))/2)
(a^3 - b^3) = 79.32212806

Answer 10

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

(a-b)^2 = (a+b)^2 - 4ab
So (a-b)^2 = 25-12 = 13
(a-b)=+-sqrt(13)

Plug in values to get the answer. However, double check it, since one of the two values of (a-b) might be wrong.