Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y²+3x+5y+9/4=0

검증된 단계별 안내

Start with the given equation: \(x^{2} + y^{2} + 3x + 5y + \frac{9}{4} = 0\).

Group the \(x\) terms and \(y\) terms together and move the constant to the other side: \(x^{2} + 3x + y^{2} + 5y = -\frac{9}{4}\).

Complete the square for the \(x\) terms: take half of the coefficient of \(x\) (which is \(3\)), square it, and add it inside the equation. Half of \(3\) is \(\frac{3}{2}\), and its square is \(\left(\frac{3}{2}\right)^{2} = \frac{9}{4}\).

Complete the square for the \(y\) terms: take half of the coefficient of \(y\) (which is \(5\)), square it, and add it inside the equation. Half of \(5\) is \(\frac{5}{2}\), and its square is \(\left(\frac{5}{2}\right)^{2} = \frac{25}{4}\).

Add these squares to both sides of the equation to keep it balanced: \(x^{2} + 3x + \frac{9}{4} + y^{2} + 5y + \frac{25}{4} = -\frac{9}{4} + \frac{9}{4} + \frac{25}{4}\), then rewrite the left side as perfect square trinomials and simplify the right side.

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주요 개념

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Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x + p)² = q. It involves adding and subtracting a constant to create a perfect square trinomial, which simplifies solving or rewriting equations, especially for conic sections like circles.

Standard Form of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Converting the general form to this form helps identify the circle's key features and makes graphing straightforward.

Identifying the Center and Radius

Once the equation is in standard form, the center of the circle is given by the coordinates (h, k), and the radius is the square root of the constant on the right side. This information is essential for graphing and understanding the circle's position and size.