Textbook Question
Identify each equation without completing the square.
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 8, Problem 49
Identify each equation without completing the square. y2 - 4x + 2y + 21 = 0
Verified step by step guidance1
Rewrite the given equation to group the terms involving \( y \) together and isolate the \( x \) terms: \( y^2 + 2y - 4x + 21 = 0 \).
Move the \( x \) terms and constant to the other side to focus on the \( y \) terms: \( y^2 + 2y = 4x - 21 \).
Recognize that the equation is quadratic in \( y \) and linear in \( x \), which suggests it might represent a parabola that opens horizontally.
Recall the standard form of a parabola that opens left or right is \( (y - k)^2 = 4p(x - h) \), where \( (h, k) \) is the vertex.
Based on the structure and terms, identify the conic as a parabola without completing the square.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Identifying Conic Sections
Conic sections are curves obtained by intersecting a plane with a cone, including circles, ellipses, parabolas, and hyperbolas. Each conic has a standard form equation involving x and y variables. Recognizing the type of conic from its general equation is essential before further manipulation.
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General Form of Conic Equations
The general form of a conic equation is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. By analyzing the coefficients, especially those of x^2 and y^2, one can determine the conic type. For example, if only one variable is squared, the conic is a parabola.
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Completing the Square (Conceptual Understanding)
Completing the square is a method to rewrite quadratic expressions in a form that reveals the conic's center and shape. Although the question asks to identify the conic without completing the square, understanding this process helps in recognizing the conic type from the equation's structure.
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Related Practice
Textbook Question
Graph each ellipse and give the location of its foci. 9(x − 1)²+4(y+3)² = 36
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Textbook Question
Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. x2 + 6x - 4y + 1 = 0
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Textbook Question
Graph each ellipse and give the location of its foci. (x − 1)²/2 + (y +3)² /5= 1
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Textbook Question
In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes.
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Textbook Question
In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes.
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