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Completing the square is a handy algebraic technique used to simplify quadratic expressions, making it easier to work with equations, particularly when dealing with circles. The process involves manipulating the equation to achieve a perfect square trinomial. This transformation allows us to express something like

• \(ax^2 + bx + c\) in the form \((x-h)^2 = c\).

For a given quadratic term such as \(x^2 - 4x\), completing the square goes as follows:1. Take the coefficient of \(x\), which is -4.2. Divide by 2: \(-4 / 2 = -2\).3. Square the result: \((-2)^2 = 4\).4. Add and subtract 4 in the equation to form a complete square: \(x^2 - 4x + 4 = (x - 2)^2\).In our problem, we use these steps for both \(x\) and \(y\) terms to convert the original circle equation into a format that reveals key details about the circle's properties. This makes working with circle equations more manageable.

The standard form of a circle's equation is an essential tool in geometry, allowing us to easily identify the circle's fundamental properties. The equation is generally expressed as:\[ (x - h)^2 + (y - k)^2 = r^2 \]Here:- \((h, k)\) is the center of the circle.- \(r\) represents the radius.This form is particularly straightforward because it directly reveals both the position and the size of the circle.In the problem we are working on, we aimed to manipulate the original equation \(x^2 + y^2 - 4x + 6y - 36 = 0\) and use completing the square to convert it into this form. Once achieved, solutions become intuitive. The hidden features of the circle, like its center and radius, become apparent with this neat and compact expression.

Finding the center of a circle from its equation is crucial, particularly when analyzing or graphing the circle. Once we have put the equation into the standard form, locating the center becomes much simpler. Look for variables inside parentheses:- In \((x - h)^2 + (y - k)^2 = r^2\), the values of \(h\) and \(k\) pinpoint the center.- The signs are key; \(x-h\) indicates \(x\) has shifted right by \(h\) unless \(h\) is negative, in which case it's a left shift.In our circle equation \((x - 2)^2 + (y + 3)^2 = 49\), we deduce:

• The \(x\)-coordinate centers at \(h = 2\).
• The \(y\)-coordinate centers at \(k = -3\), due to the equation format.

Thus, the center of the circle is at \((2, -3)\). Understanding this allows you to not only interpret the equation but also graph the circle accurately.

The radius of a circle is a fundamental geometric property—essentially the distance from the center to any point on the circle. Recognizing the radius from the standard form of a circle's equation is straightforward:- In the form \((x - h)^2 + (y - k)^2 = r^2\), \(r^2\) determines the radius.- Solving for \(r\) simply involves taking the square root of this term.In the final version of our circle equation, \((x - 2)^2 + (y + 3)^2 = 49\), it is clear that:

• \(r^2 = 49\), implying the radius \(r = \sqrt{49} = 7\).

By identifying \(r\), you gain insight into the size of the circle, a crucial part of visualizing and graphing it. An accurate computation here ensures you can further analyze or work with the circle in various applications.