if a+b=1 and a squared + b squared=2
what does a to the 3rd + b to the 3rd equal?
If b is 1, b cubed is one.
That would work for the first half, but doesn't extend to a to the 3rd+b to the 3rd and what weould 2 squard be? Thank you so much for helping
Are you saying:
if a+b=1 and if a^2 + b^2 =2 find a^3 + b^3 ??
if so, then you have to play with the expansion of (a+b)^??
(a+b)^2 = a^2 + 2ab + b^2 subbing in your given
1 = 2 + 2ab ------> ab = -1/2
then (a+b)^3 = a^3 + 3(a^2)b + 3a(b^2) + b^3
(a+b)^3 = a^3 + b^3 + 3ab(a + b)
1 = a^3 + b^3 + 3(-1/2)(1)
a^3 + b^3 = -3/2 - 1
= -5/2
oops on the last two lines
should have been
a^3 + b^3 = 3/2 + 1
= 5/2
THANK YOU!!
To solve for a^3 + b^3, we need to use the expansion of (a+b)^3.
First, let's solve for the value of ab using the given equations.
Given: a + b = 1 ...(1)
a^2 + b^2 = 2 ...(2)
To find the value of ab, we can square equation (1) and subtract equation (2) from it.
Squaring equation (1): (a + b)^2 = 1^2
a^2 + 2ab + b^2 = 1
Subtracting equation (2) from the result:
(a^2 + 2ab + b^2) - (a^2 + b^2) = 1 - 2
2ab = -1
ab = -1/2
Now, let's find the value of a^3 + b^3 using the expanded form of (a+b)^3.
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Substituting the value of ab = -1/2 into the equation:
(a+b)^3 = a^3 + 3(a^2)(-1/2) + 3a(-1/2)^2 + b^3
= a^3 - (3/2)a^2 - (3/4)a + b^3
We know that a + b = 1 from equation (1). So, b = 1 - a.
Substituting b = 1 - a into the equation:
(1)^3 = a^3 - (3/2)a^2 - (3/4)a + (1-a)^3
1 = a^3 - (3/2)a^2 - (3/4)a + (1 - 3a + 3a^2 - a^3)
1 = a^3 - a^3 - (3/2)a^2 + 3a^2 + (1 - 3a - (3/4)a)
1 = -(3/2)a^2 + (9/4)a + 1 - (9/4)a
1 = -(3/2)a^2 + (9/4)a - (9/4)a + 1
1 = -(3/2)a^2
Now, we can solve for a^3:
(3/2)a^2 = 1
a^2 = 2/3
a = ±√(2/3)
Substituting the value of a into equation (1) to solve for b:
√(2/3) + b = 1
b = 1 - √(2/3)
Finally, we can calculate a^3 + b^3:
a^3 + b^3 = (√(2/3))^3 + (1 - √(2/3))^3
= 2/3 + (1 - 2√(2/3) + 2/3)
= 2/3 + 2/3 - 2√(2/3)
= 4/3 - 2√(2/3)
So, a^3 + b^3 is equal to 4/3 - 2√(2/3).