if the equation given is y^2-6y=24x+3, how do i out this equation into standard form, using complete the square as needed?
why complete the square?
24x+3 = y^2-6y
24x = y^2-6y-3
x = 1/24 y^2 - 1/4 y - 1/8
That's standard form. To complete the square, you'd do
y^2-6y = 24x+3
y^2-6y+9 = 24x+3+9
(y-3)^2 = 24x+12
24x = (y-3)^2 - 12
x = 1/24 (y-3)^2 - 1/2
Looks like you need to review your text to get an idea of the mechanics involved.
This is also unusual, in that the axis of the parabola is horizontal, rather than vertical.
To put the given equation, y^2 - 6y = 24x + 3, into standard form by completing the square, follow these steps:
Step 1: Group the y-terms together on one side of the equation and move the constant term to the other side.
y^2 - 6y - 24x = 3
Step 2: To complete the square, take half of the coefficient of the y-term (-6) and square it: (-6/2)^2 = 9. Add this value to both sides of the equation.
y^2 - 6y + 9 - 24x = 3 + 9
Simplifying:
(y - 3)^2 - 24x = 12
Step 3: Move the constant term to the other side of the equation.
(y - 3)^2 = 24x + 12
Step 4: Divide both sides of the equation by any common factor to make the coefficient of the x-term 1. In this case, we can divide by 12.
(y - 3)^2/12 = (24x + 12)/12
Simplifying:
(y - 3)^2/12 = 2x + 1
Now, the equation is in standard form: (y - 3)^2/12 = 2x + 1.
To put the given equation in standard form, you will need to complete the square to isolate the variables on one side and have a constant term on the other side. Here's how you can do it:
Step 1: Start by moving the constant term (3) to the right side of the equation:
y^2 - 6y - 24x = 3
Step 2: Rearrange the equation to have the variables (y) on the left side and the constant (24x + 3) on the right side:
(y^2 - 6y) = 24x + 3
Step 3: To complete the square, take half of the coefficient of the y-term, square it, and add it to both sides of the equation. In this case, the coefficient of the y-term is -6, so:
(y^2 - 6y + 9) = 24x + 3 + 9
Now, the left side of the equation can be factored as the square of a binomial:
(y - 3)^2 = 24x + 12
Step 4: Finally, move the constant term (12) to the right side of the equation:
(y - 3)^2 - 12 = 24x
The equation is now in standard form, where the variables are isolated on the left side and the constant term is on the right side.
Standard Form: (y - 3)^2 - 12 = 24x