Question

What is the standard form of 5x^2+5y^2-10x+20y-30=0

Answer 1

$5x^2+5y^2-10x+20y-30=0$

We need to get the equation in square form

We apply completing the square method

$5x^2-10x+5y^2+20y-30=0$

Divide all the terms by 5

$x^2-2x+y^2+4y-6=0$

Group x  terms and y terms separately  

$(x^2-2x)+(y^2+4y)-6=0$

By completing the square method, we take half of coefficient of x  and square it

$\frac{-2}{2} = -1$

square it $(-1)^2= 1$

Add 1 on both sides

$(x^2-2x+1)+(y^2+4y)-6=+1$

$\frac{4}{2} = 2$

square it $(2)^2= 4$

Add 4 on both sides

$(x^2-2x+1)+(y^2+4y+4)-6=0+1+4$

Add 6 on both sides

$(x^2-2x+1)+(y^2+4y+4)=6+1+4$

Now factor both parenthesis

$(x-1)^2+(y+2)^2=11$

The equation of the circle is $(x-1)^2+(y+2)^2=11$