Question

How do you solve using completing the square method `-3x^2+30x-74`?

Answer 1

This cannot really be "solved" because it is not an equation and it is not "equal" to anything. It can only be written in a different form.

Answer 2

I got:

`y = -3(x - 5)^2 + 1`

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You have:

`y = -3x^2 + 30x - 74`

What you can start with is to make the `x^2` term have a perfect-square coefficient (`1` is fine). That makes it easier to come up with a factor.

Then, halve the `x` term's coefficient and square it, adding it to both sides.

`y/3 = -x^2 + 10x - 74/3` (divide by 3)

`-y/3 = x^2 - 10x + 74/3` (switch signs)

Half of `-10` is `-5`, squared is `+25`. To make common denominators happen, `25 = 75/3`. Now add it to both sides.

`=> -y/3 + 75/3 = x^2 - 10x + 74/3 + 75/3`

`=> -y/3 = x^2 - 10x + 75/3 - 1/3`

`=> -y/3 = color(green)(x^2 - 10x + 25) - 1/3`

`=> -y/3 = color(green)((x - 5)^2) - 1/3`

Now you can return it back to `y` by reversing the two things we did in the first steps:

`=> y/3 = -(x - 5)^2 + 1/3`

`=> color(blue)(y = -3(x - 5)^2 + 1)`