we know the discriminant d^2 = 961-28c
so, the roots are
(-31 + √(961-28c))/14
and
(-31 - √(961-28c))/14
since they are reciprocals,
(-31 - √(961-28c))/14 = 14/(-31 + √(961-28c))
961-(961-28c) = 196
c = 7
so, the roots are
(-31 + √(961-28c))/14
and
(-31 - √(961-28c))/14
since they are reciprocals,
(-31 - √(961-28c))/14 = 14/(-31 + √(961-28c))
961-(961-28c) = 196
c = 7
A polynomial of the form ax^2 + bx + c = 0 can be solved using the quadratic formula: (-b ± √(b^2 - 4ac)) / (2a).
In this case, the polynomial f(x) = 7x^2 + 31x + c = 0. So, a = 7, b = 31, and c is the value we are trying to find.
Since the roots are reciprocals of each other, let's assume the roots are r and 1/r.
Applying the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
x = (-31 ± √(31^2 - 4 * 7 * c)) / (2 * 7)
The roots of f(x) are the values of x that make f(x) equal to zero. Therefore, we can set the discriminant (b^2 - 4ac) to zero, as we want to find the value of c that satisfies this condition.
b^2 - 4ac = 0
(31^2) - 4(7)(c) = 0
961 - 28c = 0
28c = 961
c = 961 / 28
So, the value of c is 961 / 28.