85th Problem 2016

Algebra Level 2

Write the vertex of the following function as ( x , y ) \left( x,y \right) :

f ( x ) = − 2 x 2 − 12 x − 21 f\left( x \right) ={ -2x }^{ 2 }-12x-21

Find x + y x+y

Check out the set: 2016 Problems


The answer is -6.

Angela Fajardo by Angela Fajardo , 19, Philippines

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1 solution

Hung Woei Neoh
Jun 6, 2016

There are 3 3 methods to find the vertex of quadratic functions f ( x ) = a x 2 + b x + c f(x) = ax^2 + bx + c

  1. The coordinates of the vertex is defined as ( − b 2 a , f ( − b 2 a ) ) \left(-\dfrac{b}{2a},f\left(-\dfrac{b}{2a}\right)\right)
  2. Complete the square and write in the form f ( x ) = a ( x − p ) 2 + q f(x) = a(x-p)^2 + q . The vertex is given as ( p , q ) (p,q)
  3. Use Calculus. At vertexes, f ′ ( x ) = 0 f'(x) = 0

For this question, I will demonstrate Method 2

f ( x ) = − 2 x 2 − 12 x − 21 = − 2 ( x 2 + 6 x + ( 6 2 ) 2 − ( 6 2 ) 2 ) − 21 = − 2 ( ( x + 3 ) 2 − 9 ) − 21 = − 2 ( x + 3 ) 2 + 18 − 21 = − 2 ( x + 3 ) 2 − 3 f(x) = -2x^2 - 12x-21\\ =-2(x^2 + 6x +\left(\dfrac{6}{2}\right)^2 - \left(\dfrac{6}{2}\right)^2 ) - 21\\ =-2\left((x+3)^2 - 9\right) - 21\\ =-2(x+3)^2+ 18-21\\ =-2(x+3)^2 - 3

The vertex is ( − 3 , − 3 ) (-3,-3)

x = − 3 , y = − 3 , x + y = ( − 3 ) + ( − 3 ) = − 6 x=-3,\;y=-3,\;x+y=(-3)+(-3) = \boxed{-6}

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