a+b=m, ab=3m+6, and a^2+b^2=100

What is M?

M = 2497

To solve this problem, we can use a system of equations to find the values of 'a' and 'b'.

First, let's look at the given equations:

1. a + b = m
2. ab = 3m + 6
3. a^2 + b^2 = 100

To find 'a' and 'b', we'll solve these equations step by step:

Step 1: Solve equation 1 for 'a' or 'b':
Since the first equation doesn't have a coefficient for 'a' or 'b', it's easier to solve for 'a' or 'b'. Let's solve for 'a':
a = m - b

Step 2: Substitute the expression for 'a' in equation 2:
Now, substitute the value of 'a' from step 1 into equation 2:
(m - b) * b = 3m + 6

Step 3: Simplify the equation:
Expand the left side of the equation by multiplying:
b^2 - mb = 3m + 6

Step 4: Rearrange equation 3 and substitute the expression from step 1:
Rearrange equation 3 to isolate 'b':
b^2 + 2mb + (m^2 - 100) = 0

Step 5: Solve the quadratic equation for 'b':
Now we have a quadratic equation in terms of 'b'. We can solve it using the quadratic formula:
b = (-2mb ± √((2mb)^2 - 4(m^2 - 100))) / 2

Simplifying this expression will give you two possible values for 'b'.

Step 6: Substitute the value of 'b' back into equation 1:
After finding the possible values of 'b', substitute each value back into equation 1:
a = m - b

This will give you the corresponding values of 'a'.

By following these steps, you can find the possible values of 'a' and 'b' that satisfy all three equations.