What is the vertex form of #y=5x^2-30x+49#?

2 Answers
Jul 25, 2017

See a solution process below:

Explanation:

To convert a quadratic from #y = ax^2 + bx + c# form to vertex form, #y = a(x - color(red)(h))^2+ color(blue)(k)#, you use the process of completing the square.

First, we must isolate the #x# terms:

#y - color(red)(49) = 5x^2 - 30x + 49 - color(red)(49)#

#y - 49 = 5x^2 - 30x#

We need a leading coefficient of #1# for completing the square, so factor out the current leading coefficient of 2.

#y - 49 = 5(x^2 - 6x)#

Next, we need to add the correct number to both sides of the equation to create a perfect square. However, because the number will be placed inside the parenthesis on the right side we must factor it by #2# on the left side of the equation. This is the coefficient we factored out in the previous step.

#y - 49 + (5 * ?) = 5(x^2 - 6x + ?)# <- Hint: #6/2 = 3#; #3 * 3 = 9#

#y - 49 + (5 * 9) = 5(x^2 - 6x + 9)#

#y - 49 + 45 = 5(x^2 - 6x + 9)#

#y - 4 = 5(x^2 - 6x + 9)#

Then, we need to create the square on the right hand side of the equation:

#y - 4 = 5(x - 3)^2#

Now, isolate the #y# term:

#y - 4 + color(blue)(4) = 5(x - 3)^2 + color(blue)(4)#

#y - 0 = 5(x - 3)^2 + color(blue)(4)#

#y - 0 = 5(x - color(red)(3))^2 + color(blue)(4)#

The vertex is: #(3, 4)#

Jul 25, 2017

#y = 5(x - 3) + 4#

Explanation:

#y = 5x^2 - 30x + 49#
x-coordinate of vertex:
#x = -b/(2a) = 30/10 = 3#
y-coordinate of vertex:
#y(3) = 5(9) - 30(3) + 49 = 4#
Vertex (3, 4)
Vertex form of y:
#y = 5(x - 3)^2 + 4#