Question

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

Answer 1

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)`

Answer 2

`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`