Two consecutive even numbers are n and n+2 write down an algebraic for the sum of their squares. Find the number of the sum of the squares is 164
The algebraic expression for the sum of the squares of two consecutive even numbers is:
n^2 + (n+2)^2.
We want to find the value of n for which the sum of the squares is 164:
n^2 + (n+2)^2 = 164.
Expanding the expression:
n^2 + n^2 + 4n + 4 = 164.
Combining like terms:
2n^2 + 4n + 4 = 164.
Subtracting 164 from both sides:
2n^2 + 4n - 160 = 0.
Dividing both sides by 2:
n^2 + 2n - 80 = 0.
Factoring the quadratic equation:
(n + 10)(n - 8) = 0.
Setting each factor equal to zero and solving for n:
n + 10 = 0 -> n = -10,
n - 8 = 0 -> n = 8.
Since the numbers are consecutive even numbers, n cannot be -10. Therefore, the value of n is 8.
Thus, the sum of the squares of the two numbers is:
8^2 + (8+2)^2 = 64 + 100 = 164.
So, the sum of the squares is 164.