Question

If f \((5+x) = f (5-x)\) for every real x, and \(f(x)=0\) has four distinct real roots, then the sum of these roots is

• A. 0
• B. 40
• C. 10
• D. 20

Answer 1

The Correct Option is D

The function's property is given as \(f(5+x) = f(5-x)\) for all real \(x\).

This indicates symmetry around the line \(x = 5\). Consequently, if \(a\) is a root, \(10-a\) must also be a root.

Given that \(f(x)=0\) has four distinct real roots, denoted \(r_1, r_2, r_3, r_4\). Their symmetric pairings are:

• \(r_1 + (10 - r_1) = 10\)
• \(r_2 + (10 - r_2) = 10\)

Each symmetric pair of roots sums to 10. Therefore, the sum of all four roots is:

\(10 + 10 = 20\).

The sum of the four distinct real roots equals 20.