college algebra word problem

Question

Rewrite the given equation in standard form.

x^² − 6x − 24y + 105 = 0

Answer 1

Hello, thank you for taking the time to post your question!

To rewrite this expression in standard form, you want to get it into the format

(x – h)^2 = 4p(y – k)

So to start I would group the x terms together then complete the square for the x term and then simplify the expression!

x^2 – 6x = 24y – 105

(x^2 – 6x + 9) = 24y – 105 + 9

(x – 3)^2 = 24y – 96

(x – 3)^2 = 24(y – 4), which is the standard from for the given equation

I hope that helps you get moving in a better direction on this type of question! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

y = (1/24)(x-3)^2 + 4

complete the square and solve for y

in reverse, multiply both sides by 24, expand the squared term

24y = x^2 -6x + 9 + 96 = x^2 -6x + 105

vertex or minimum point is (3,4)

It's an upward opening parabola. Very flat though, given the (1/24) coefficient of the squared term