91Ó°ÊÓ

Conic sections are the curves obtained by intersecting a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas, each with unique properties and equations. The equation in our exercise, \(9 x^{2}+4 y^{2}-36 x+8 y+31=0\), represents a conic section.

By completing the square, we can transform this equation into a standard form that clearly identifies the type of conic section it represents. This process involves algebraic manipulation to reveal the shape's center, orientation, and size.

The completed square form, \(9(x-2)^2 + 4(y+1)^2 = 5\), tells us we're dealing with an ellipse since both variables are squared and have positive coefficients. Understanding conic sections and their standard equations is essential in geometry and various applications, such as satellite dish design and planetary orbits.

The standard form of a conic section's equation is crucial as it provides a clear and concise description of the shape. For an ellipse, the standard form is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \(h, k\) is the center, and \(a, b\) are the lengths of the semi-major and semi-minor axes, respectively.

In our exercise, \(9(x-2)^2 + 4(y+1)^2 = 5\) is close to this form, but we need to divide everything by 5 to match exactly. It's a straightforward method that allows us to immediately identify and analyze the properties of the conic section in question. Whenever you're given a conic section equation, getting it into the standard form should be your first approach.

Quadratic equations are fundamental to algebra and appear as parts of conic section equations. They have the general form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants, and \(a\) is not zero.

In our given problem, \(9 x^{2}-36 x+4 y^{2}+8 y\) includes two quadratic expressions - one in \(x\) and one in \(y\). By completing the square, we bring these quadratic parts into a form that reveals the maximum or minimum point, which becomes the vertex of the parabola in the context of a conic section, such as vertex-form for parabolas or center-form for ellipses and hyperbolas.

Understanding how to manipulate quadratic expressions is key to mastering concepts in algebra, thereby unlocking the ability to analyze various mathematical and real-world phenomena.

Algebraic manipulation is the process of rearranging and simplifying mathematical expressions and equations through various operations. It involves techniques such as factoring, expanding, and completing the square - as shown in the step-by-step solution to our exercise.

When we took the expression \(9 x^{2}-36 x+4 y^{2}+8 y= -31\) and performed algebraic manipulation, we were able to move from a complex equation to a more understandable equation representing an ellipse. By recognizing patterns and applying these manipulation techniques, students can solve a wide range of algebraic problems, simplify complex expressions, and reveal the underlying structure of mathematical expressions or equations.