If the sum of two numbers is 1 and the sum of the cubes of the numbers is 3, what is the sum of the squares of the numbers?
m+n=1
m^3+n^3=1
What if one number is zero, so the other number is ....
No, the sum of the cubes is 3 so its m+n=1 m^3+n^3=3
oooppppssss
m+n=1
m^3+n^3=3
If you factor the second equation..
(M+n)(M^2-mn+n^2) the first term is 1, so
(m^2+n^2 -mn)=3 equation1
but if we square the first equation given
m^2+2mn+n^2 = 1
m^2+n^2+2mn=1 equation2
subtracting equation2 from equation1
-mn-2mn=2
or -3mn=2
OK, this is not making sense. Check my work, I will try again later.
To find the sum of the squares of the two numbers, let's represent the numbers as variables. Let's say the first number is x and the second number is y.
According to the problem statement, we know two things:
1. The sum of the two numbers is 1: x + y = 1.
2. The sum of the cubes of the two numbers is 3: x^3 + y^3 = 3.
To find the sum of the squares of the two numbers, we need to find the value of x^2 + y^2.
To solve the given system of equations, we can use an algebraic method known as substitution.
Step 1: Solve Equation 1 for one variable:
x + y = 1
⇒ x = 1 - y
Step 2: Substitute the value of x in Equation 2:
(1 - y)^3 + y^3 = 3
⇒ (1 - y)(1 - y)(1 - y) + y^3 = 3
⇒ (1 - 2y + y^2)(1 - y) + y^3 = 3
⇒ (1 - 2y + y^2 - y + 2y^2 - y^3) + y^3 = 3
⇒ 1 - 2y + y^2 - y + 2y^2 - y^3 + y^3 = 3
⇒ 1 - 2y + y^2 - y + 2y^2 = 3
⇒ 3y^2 - 3y - 2 = 0
Step 3: Solve the quadratic equation:
We can solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's solve it using the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 3, b = -3, and c = -2.
y = (-(-3) ± √((-3)^2 - 4(3)(-2))) / (2(3))
⇒ (3 ± √(9 + 24)) / 6
⇒ (3 ± √33) / 6
Therefore, we have two possible values for y:
y1 = (3 + √33) / 6
y2 = (3 - √33) / 6
Step 4: Substitute the values of y into Equation 1 to find the corresponding values of x:
For y1:
x1 = 1 - y1
For y2:
x2 = 1 - y2
Now, we have two sets of values: (x1, y1) and (x2, y2).
Step 5: Calculate the sum of squares for each set of values:
For (x1, y1):
Sum of squares = x1^2 + y1^2
For (x2, y2):
Sum of squares = x2^2 + y2^2
Calculating these sums of squares will give us the answer to the problem.