Write the equation in standard parabola form?
x^2+ 4xy +4y^2 +5root5 y+ 5 = 0
To write the equation in standard parabola form, we need to eliminate the xy term by completing the square.
Given the equation x^2 + 4xy + 4y^2 + 5√5y + 5 = 0, we can follow these steps:
Step 1: Rearrange the equation to group the x^2, xy, and y^2 terms together:
(x^2 + 4xy + 4y^2) + 5√5y + 5 = 0
Step 2: Factor the x^2, xy, and y^2 terms:
(x + 2y)^2 + 5√5y + 5 = 0
Step 3: To complete the square, we need to add and subtract (4/2)^2 = 4 to the grouped term (x + 2y)^2. This ensures that the equation remains balanced:
(x + 2y)^2 + 5√5y + 5 - 4 = 0
Step 4: Simplify the equation after adding and subtracting 4:
(x + 2y)^2 + 5√5y + 1 = 0
Step 5: Rewriting the equation in standard parabola form:
(x + 2y)^2 = -5√5y - 1
The equation (x + 2y)^2 = -5√5y - 1 is now in standard parabola form.