Unknown Functions

Algebra Level 4

The function f ( x ) f(x) satisfies f ( 2 + x ) = f ( 2 − x ) f(2 + x) = f(2 - x) for all real numbers x x . If the equation f ( x ) = 0 f(x) = 0 has exactly four distinct real roots, find the sum of these roots.

This is not original.


The answer is 8.

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Jason Hughes by Jason Hughes , 25, USA

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2 solutions

Trevor Arashiro
Feb 7, 2015

Substitute x = a − 2 x=a-2 we get

f ( a ) = f ( 4 − a ) f(a)=f(4-a)

This if x is a solution, 4-x is a solution.

Now, x represents all values of x possible. Let's substitute p and q for two SPECIFIC distinct values of the 4 possible roots of this function.

We have the solutions p , 4 − p , q , 4 − q p,4-p,q,4-q

If we sum these we get

4 − p + p + 4 − q + q = 8 4-p+p+4-q+q=8

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Shahzad Karim
Feb 9, 2015

This function is symmetric about x=2. Thus if ( 'a' measured from line of symmetry = 2+ a if measured from origin) is a solution then ( '-a' measured from line of symmetry = 2-a if measured from origin) is also a solution. Similarly for b. Therefore, sum of roots = 2 + a + 2 - a + 2 + b + 2 - b = 8

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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