Step-by-step explanation:
Let's solve the problem step by step:
Let's assume the first number is x.
According to the given information, the second number is 4 more than x, which can be represented as (x + 4).
The sum of the squares of the two numbers is 72, so we can write the equation as:
x^2 + (x + 4)^2 = 72
Expanding and simplifying the equation:
x^2 + (x^2 + 8x + 16) = 72
2x^2 + 8x + 16 = 72
2x^2 + 8x - 56 = 0
Dividing the equation by 2 to simplify further:
x^2 + 4x - 28 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 4, and c = -28. Substituting these values into the quadratic formula:
x = (-4 ± √(4^2 - 4(1)(-28))) / (2(1))
x = (-4 ± √(16 + 112)) / 2
x = (-4 ± √128) / 2
x = (-4 ± 8√2) / 2
Simplifying further:
x = -2 ± 4√2
Therefore, the two possible values for x are:
x = -2 + 4√2
x = -2 - 4√2
Since x represents a positive real number, we can discard the negative value:
x = -2 + 4√2
To find the second number, we can substitute this value back into x + 4:
Second number = (-2 + 4√2) + 4
Second number = 2 + 4√2
So, the numbers are:
First number = -2 + 4√2
Second number = 2 + 4√2
