x is a perfect square and x+5 is also a perfect square and x-5 is a perfect square.what is the value of x?
To find the value of x, we need to solve the given conditions. Let's break down the three conditions given:
1. x is a perfect square.
2. x + 5 is a perfect square.
3. x - 5 is a perfect square.
Let's denote the perfect squares as follows:
1. x = a^2
2. x + 5 = b^2
3. x - 5 = c^2
Now, let's try to eliminate the variable x from equations (2) and (3) to solve for b^2 and c^2:
Adding equations (2) and (3):
(b^2) + (c^2) = (x + 5) + (x - 5)
(b^2) + (c^2) = 2x
Substituting x = a^2 from equation (1):
(b^2) + (c^2) = 2(a^2)
Given that a, b, and c are integers, we need to find a value of "a" that satisfies the equation.
Now, let's consider the possible values of (b^2) and (c^2) that satisfy the above equation:
1 + 1 = 2 --> b = 1, c = 1
4 + 4 = 8 --> b = 2, c = 2
9 + 9 = 18 --> b = 3, c = 3
16 + 16 = 32 --> b = 4, c = 4
25 + 25 = 50 --> b = 5, c = 5
Since we need to find the value of x, we can substitute the values of b and c in equations (2) and (3) respectively:
For b = c = 1:
x + 5 = (1^2)
x = 1 - 5
x = -4
For b = c = 2:
x + 5 = (2^2)
x = 4 - 5
x = -1
For b = c = 3:
x + 5 = (3^2)
x = 9 - 5
x = 4
For b = c = 4:
x + 5 = (4^2)
x = 16 - 5
x = 11
For b = c = 5:
x + 5 = (5^2)
x = 25 - 5
x = 20
Therefore, the values of x that satisfy the given conditions are x = -4, -1, 4, 11, and 20.