Write the equation in standard form for the circle $x^2+y^2+10y-25= 0$ .

Full solution

Q. Write the equation in standard form for the circle

x^2+y^2+10y-25= 0

Identify equation and need: Identify the given equation and the need to complete the square for $y$ . $\newline$ The given equation is $x^2 + y^2 + 10y - 25 = 0$ . To write the equation in standard form, we need to complete the square for the $y$ terms.
Group and leave space: Group the $y$ terms and leave a space for completing the square. $x^2 + (y^2 + 10y + \underline{\hspace{1em}}) - 25 = 0$ We will add and subtract the same value inside the parentheses to complete the square.
Calculate value for completion: Calculate the value needed to complete the square for $y$ . $\newline$ To complete the square, we take half of the coefficient of $y$ , which is $\frac{10}{2} = 5$ , and square it, resulting in $5^2 = 25$ .
Add and subtract calculated value: Add and subtract the calculated value inside the parentheses. $\newline$ $x^2 + (y^2 + 10y + 25 - 25) - 25 = 0$ $\newline$ Now we have the perfect square trinomial for $y$ and an extra $-25$ that we will move outside the parentheses.
Write perfect square trinomial: Write the perfect square trinomial and simplify the equation. $\newline$ $x^2 + (y + 5)^2 - 25 - 25 = 0$ $\newline$ Combine the constant terms on the right side.
Simplify equation: Combine the constants and write the equation in standard form. $\newline$ $x^2 + (y + 5)^2 - 50 = 0$ $\newline$ Add $50$ to both sides to get the standard form. $\newline$ $x^2 + (y + 5)^2 = 50$