82nd Problem 2016

Algebra Level 2

Write the vertex of the following function as ( X , Y ) \left( X,Y \right) :

f ( x ) = x 2 − 6 x + 5 f\left( x \right) ={ x }^{ 2 }-6x+5

Find X + Y X+Y

Check out the set: 2016 Problems


The answer is -1.

Angela Fajardo by Angela Fajardo , 19, Philippines

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1 solution

Hung Woei Neoh
May 30, 2016

There are 3 methods to do this:

  1. Formula for vertex of quadratic functions: ( − b 2 a , f ( − b 2 a ) ) \left(-\dfrac{b}{2a},f(-\dfrac{b}{2a})\right)

  2. Completing the square: f ( x ) = a ( x − p ) 2 + q f(x) = a(x-p)^2 + q , and the vertex is ( p , q ) (p,q)

  3. Use calculus. At vertexes, f ′ ( x ) = 0 f'(x) = 0

For this question, I will show method 2 2 :

f ( x ) = x 2 − 6 x + 5 = x 2 − 6 x + ( − 6 2 ) 2 − ( − 6 2 ) 2 + 5 = ( x − 3 ) 2 − 9 + 5 = ( x − 3 ) 2 − 4 f(x) = x^2-6x+5\\ =x^2-6x+\left(\dfrac{-6}{2}\right)^2 -\left(\dfrac{-6}{2}\right)^2 + 5\\ =(x-3)^2 - 9 + 5\\ =(x-3)^2-4

The vertex is: ( 3 , − 4 ) (3,-4)

X = 3 , Y = − 4 , X + Y = 3 + ( − 4 ) = − 1 X=3,\;Y=-4,\;X+Y = 3+(-4) = \boxed{-1}

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