he sum of two numbers is 15 the sum of the squares of the same two numbers is 172 fined the difference between the two numbers
To find the difference between the two numbers, let's assign variables to the numbers. Let's say the first number is "x" and the second number is "y".
From the information given, we have two equations:
1) x + y = 15 (sum of the two numbers is 15)
2) x^2 + y^2 = 172 (sum of the squares of the two numbers is 172)
We can solve this system of equations to find the values of x and y.
There are several methods to solve this system, such as substitution or elimination. Let's use the elimination method:
Multiply equation 1 by 2:
2(x + y) = 2(15)
2x + 2y = 30
Rearrange the equation:
2x = 30 - 2y
x = 15 - y
Substitute this value of x into equation 2:
(15 - y)^2 + y^2 = 172
225 - 30y + y^2 + y^2 = 172
2y^2 - 30y + 53 = 0
Now we have a quadratic equation. We can solve it using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 2, b = -30, and c = 53.
y = (-(-30) ± √((-30)^2 - 4(2)(53))) / (2(2))
y = (30 ± √(900 - 424)) / 4
y = (30 ± √476) / 4
y = (30 ± 2√119) / 4
y = (15 ± √119) / 2
Now we have two possible values for y. Let's calculate the corresponding values of x using the equation x = 15 - y:
For y = (15 + √119) / 2:
x = 15 - (15 + √119) / 2
x = (30 - (15 + √119)) / 2
x = (30 - 15 - √119) / 2
x = (15 - √119) / 2
x ≈ 0.108
For y = (15 - √119) / 2:
x = 15 - (15 - √119) / 2
x = (30 - (15 - √119)) / 2
x = (30 - 15 + √119) / 2
x = (15 + √119) / 2
x ≈ 14.892
Therefore, the two numbers are approximately 0.108 and 14.892. To find the difference between these numbers, we subtract the smaller number from the larger number:
Difference = 14.892 - 0.108
Difference ≈ 14.784
So, the difference between the two numbers is approximately 14.784.
a + b = 15
a^2 + b^2 = 172
a + b = 15
=> a = 15 - b
=> a^2 = (15-b)^2
So,
a^2 + b^2 = 172
=> (15-b)^2 + b^2 = 172
Solve for b, find the difference.