교과서 질문
Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 25x²+4y² – 150x + 32y + 189 = 0
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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
8장, 문제 56
Identify each equation without completing the square.
검증된 단계별 안내1
Start by recognizing the general form of a conic section equation. The given equation is \( y^2 + 8x + 6y + 25 = 0 \). This resembles the general form of a parabola, \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), where either \( A = 0 \) or \( C = 0 \).
In the given equation, notice that there is no \( x^2 \) term, which means \( A = 0 \). This suggests that the equation could represent a parabola that opens horizontally.
Next, identify the coefficients: \( A = 0 \), \( B = 0 \), \( C = 1 \), \( D = 8 \), \( E = 6 \), and \( F = 25 \). Since \( C \neq 0 \) and \( A = 0 \), this confirms that the equation is a parabola.
To further analyze the equation, rearrange it to isolate the \( y \) terms: \( y^2 + 6y = -8x - 25 \). This form helps in identifying the vertex and direction of the parabola.
Finally, recognize that the equation is in a form that can be transformed into the standard form of a parabola by completing the square on the \( y \) terms, but since the task is to identify without completing the square, we conclude that the equation represents a horizontally oriented parabola.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Quadratic Equations
A quadratic equation is a polynomial equation of degree two, typically expressed in the standard form ax^2 + bx + c = 0. In the given equation, y^2 + 6y + 8x + 25 = 0, the presence of the y^2 term indicates that it is a quadratic in y. Understanding the structure of quadratic equations is essential for identifying their properties and solutions.
추천 영상:
05:35
Introduction to Quadratic Equations
Conic Sections
Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The equation provided can represent different conic sections depending on its form. In this case, it can be rearranged to identify whether it represents a parabola, ellipse, or hyperbola, which is crucial for understanding the geometric implications of the equation.
추천 영상:
3:08Geometries from Conic Sections
Completing the Square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, making it easier to solve or analyze. Although the question specifies not to complete the square, understanding this technique is vital for recognizing the vertex form of a quadratic and for solving quadratic equations efficiently.
추천 영상:
06:24Solving Quadratic Equations by Completing the Square
관련 실천
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교과서 질문
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교과서 질문
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교과서 질문
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교과서 질문
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