the function with the sum of roots of 11 and product of roots is 11 is..?
thanks for the help!
Assuming the function is quadratic with an x^2 coefficient of 1,
in the form ax^2 + bx + c = 0.
x = [-b +/- sqrt(b^2 -4ac)]/2a
root sum = -b/a = -b
root product = (1/4a^2)*(-4ac) = -c
x^2 -11x -11 = 0 is the function
Roots are (1/2)(11 + sqrt165)
and (1/2)(11 - sqrt165)
There are many other functions that would meet your criteria, but all of the quadratic ones must be a multiple of the one above.
To find the function with the given conditions, we can use the fact that for a quadratic equation of the form ax^2 + bx + c = 0, the sum of the roots is given by -b/a, and the product of the roots is given by c/a.
Given that the sum of the roots is 11 and the product of the roots is also 11, we can set up the following equations:
- b/a = 11 (1)
c/a = 11 (2)
Since we have two equations with two variables (b and c), we can solve them simultaneously to find the values of b and c.
From equation (1), we can isolate b:
b = -11a (3)
Substituting equation (3) into equation (2), we get:
c/a = 11
Multiplying both sides by a, we get:
c = 11a (4)
Now we have expressions for b and c in terms of a. We can substitute these expressions into the general quadratic function: f(x) = ax^2 + bx + c.
Substituting b and c, we have:
f(x) = ax^2 - 11ax + 11a
Therefore, the function with the sum of roots of 11 and product of roots also 11 is f(x) = ax^2 - 11ax + 11a.
Note: The value of 'a' can be any non-zero real number since it does not affect the sum or product of the roots.