Question:hard

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

Updated On: Jan 15, 2026
  • 0
  • 40
  • 10
  • 20
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The Correct Option is D

Solution and Explanation

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.

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