complete the square, find vertex?

Question

complete the square of y = 5x² - 30x + 9

and find the vertex.

Answer 1

if you have a quadratic of the form Ax^2 + bx + c, you need a value of  (b/2)^2/A to complete the square.   We add and subtract this value on the left side of the equation.   In this case (b/2)^2/A = (30/2)^2/A = 45

 

y  = 5x^2 -30x + 45 - 45 + 9

             = 5(x^2 -6x + 9) - 45 + 9

              = 5(x-3)^2 - 36

 

Now that the equation is written in vertex form, it is clear that the vertex is at  (3, -36)

Answer 2

Vertex form of a quadratic equation is

 

y = a(x - h)² + k

 

where:

a is the coefficient of the leading term

(h, k) is the vertex

 

Your equation in vertex form is

 

y = 5(x² - 6x + 9) - 45 + 9

 

 

Note that the term in parenthesis is a perfect square trinomial.

 

y = 5(x - 3)(x - 3) - 36

 

y = 5(x - 3)² - 36