Equation of a Sphere (Using Completing the Square)

Houston Math Prep

31 Jan 202116:48

EducationalLearning

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TLDRThis video from the Houston Mathprep series dives into transforming the general form of a sphere equation into the center-radius form by completing the square. It explains the process of adjusting the equation to reveal the sphere's center and radius, providing step-by-step examples. The tutorial is designed to help viewers understand how to manipulate algebraic expressions to identify key geometric properties of a sphere.

Takeaways

📚 The video is part of a calculus series by Houston Mathprep, focusing on sphere equations and the method of completing the square.
🌐 The standard form of a sphere's equation is given as \((x - a)^2 + (y - b)^2 + (z - c)^2 = r^2\), with \((a, b, c)\) being the center and \(r\) the radius.
🔍 Sometimes, the equation of a sphere might not be given in the standard form but in a general form involving \(x^2 + y^2 + z^2\) with additional linear terms and a constant.
📉 The process of completing the square is used to transform the general form of a sphere's equation into the standard form to easily identify the center and radius.
🔢 Completing the square involves taking an expression and ensuring it factors into the same thing twice, by adding a specific value to both sides of the equation to maintain balance.
📝 The script demonstrates how to complete the square with examples, including handling terms with coefficients and constants, and adjusting for non-integer values.
📚 Examples are provided to illustrate the process, such as transforming the equation \(x^2 + y^2 + z^2 + 2x + 4y - 6z - 2 = 0\) into the standard form of a sphere's equation.
🧩 The script explains that after completing the square, the center of the sphere can be identified by the values opposite the signs in the completed square terms.
🔑 The radius of the sphere is found by taking the square root of the constant term on the right side of the equation once it's in the standard form.
⚙️ The video also covers cases where the coefficients of the squared terms are not one, and how to adjust the equation to fit the standard form by dividing by the common coefficient.
👍 The final takeaway is that completing the square is a valuable technique for dealing with sphere equations in calculus, making it easier to identify key properties like the center and radius.

Q & A

What is the standard center-radius form of a sphere equation?
-The standard center-radius form of a sphere equation is given by \((x - a)^2 + (y - b)^2 + (z - c)^2 = r^2\), where \((a, b, c)\) is the center of the sphere in 3D space, and \(r\) is the radius.
What is the general form of a sphere equation and how does it differ from the center-radius form?
-The general form of a sphere equation is \(Ax^2 + By^2 + Cz^2 + Dx + Ey + Fz + G = 0\), where \(A\), \(B\), and \(C\) are coefficients, and \(D\), \(E\), \(F\), and \(G\) are constants. It differs from the center-radius form as it may include linear terms and a constant, and requires completing the square to find the center and radius.
What is the process of completing the square and why is it used in sphere equations?
-Completing the square is a mathematical technique used to transform a quadratic trinomial into a perfect square trinomial plus a constant. It is used in sphere equations to convert the general form into the center-radius form, making it easier to identify the sphere's center and radius.
How do you complete the square for a term like \(x^2 + 10x\)?
-To complete the square for \(x^2 + 10x\), take half of the coefficient of \(x\) (which is 10), square it to get 25, and add it to both sides of the equation. This results in \(x^2 + 10x + 25 = (x + 5)^2\).
What is the center of the sphere given by the equation \(x^2 + y^2 + z^2 + 2x + 4y - 6z - 2 = 0\)?
-The center of the sphere is at \((-1, -2, 3)\), found by completing the square for each variable and changing the signs of the constants inside the squared terms.
What is the radius of the sphere with the equation \(x^2 + y^2 + z^2 + 2x + 4y - 6z - 2 = 0\)?
-The radius of the sphere is 4, determined by taking the square root of the constant term on the right side of the equation after completing the square and moving the -2 to the other side.
How do you handle coefficients that are not 1 in front of the squared terms when completing the square for a sphere equation?
-If the coefficients in front of the squared terms are not 1, you can divide the entire equation by the common coefficient to simplify the equation to a form where completing the square is more straightforward.
What is the center of the sphere given by the equation \(3x^2 + 3y^2 + 3z^2 + 6x + y = 0\)?
-The center of the sphere is at \((-1, -\frac{1}{6}, 0)\), found by dividing the entire equation by 3, completing the square for each variable, and changing the signs of the constants inside the squared terms.
What is the radius of the sphere with the equation \(3x^2 + 3y^2 + 3z^2 + 6x + y = 0\)?
-The radius of the sphere is \(\frac{\sqrt{37}}{6}\), determined by taking the square root of the constant term on the right side of the equation after completing the square and simplifying.
Why is it necessary to have the same coefficient for all squared terms in a sphere equation?
-Having the same coefficient for all squared terms ensures that the equation represents a sphere. If the coefficients differ, the equation may represent a different type of quadratic surface, not a sphere.
How do you rationalize the radius expression for a sphere with a fractional radius?
-To rationalize the radius expression, multiply the numerator and denominator by the conjugate of the denominator. For example, if the radius is \(\sqrt{13}/2\), multiply by \(\sqrt{2}/\sqrt{2}\) to get \(\sqrt{26}/2\).

Outlines

00:00

📚 Introduction to Sphere Equations and Completing the Square

This paragraph introduces the concept of sphere equations in calculus 3, focusing on the center-radius form and the general form. It explains the potential need to convert from the general form to the center-radius form using the method of completing the square. The importance of identifying the center and radius of a sphere from its equation is highlighted, along with a brief explanation of what completing the square entails and how it is applied to quadratic expressions.

05:01

🔍 Completing the Square for Sphere Equations

The paragraph delves deeper into the process of completing the square, especially for equations not initially presented in the center-radius form of a sphere. It outlines the steps to transform a general form equation into the center-radius form by adjusting for linear terms and constant terms. The method involves grouping like terms and adding specific values to both sides of the equation to create perfect squares, which simplifies the identification of the sphere's center and radius.

10:02

📘 Practical Examples of Completing the Square for Sphere Equations

This section provides practical examples to illustrate the process of completing the square for sphere equations. It walks through the steps of grouping terms, identifying coefficients, and adding necessary values to both sides of the equation to balance it. The examples demonstrate how to transform the general form of a sphere equation into the center-radius form, making it easier to determine the sphere's center and radius.

15:02

🔑 Finding the Center and Radius of a Sphere

The final paragraph discusses the culmination of the process, which is identifying the center and radius of a sphere from its equation. It emphasizes the importance of recognizing the signs and coefficients in the completed square form and how they directly relate to the sphere's geometric properties. The paragraph concludes with a summary of the method and an invitation to continue learning in the next video.

Mindmap

Keywords

💡Sphere Equation

A sphere equation is a mathematical representation that defines the set of all points in three-dimensional space that are equidistant from a fixed point, known as the center. In the video, the sphere equation is discussed in two forms: the center-radius form and the general form. The center-radius form is given by (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a, b, c) is the center and r is the radius. The general form includes squared terms for x, y, and z, along with possible linear terms and a constant.

💡Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into a perfect square trinomial plus or minus a constant. This method is essential in the video for converting the general form of a sphere equation into the center-radius form. The process involves adding and subtracting the square of half the coefficient of the linear term to both sides of the equation, as demonstrated with the terms x^2 + 10x and y^2 - 8y in the script.

💡Center of a Sphere

The center of a sphere refers to the point in three-dimensional space that all points on the surface of the sphere are equidistant from. It is represented by the coordinates (a, b, c) in the center-radius form of the sphere equation. In the video, finding the center involves rearranging the equation after completing the square to isolate the squared terms, which then correspond to the coordinates of the center.

💡Radius of a Sphere

The radius of a sphere is the distance from the center to any point on the surface of the sphere. It is a key parameter in the sphere equation, represented by r in the center-radius form. The radius can be determined after completing the square and is the square root of the constant term on the right side of the equation, as shown in the video when the radius is calculated from r^2 = 16.

💡General Form

The general form of a sphere equation includes squared terms for x, y, and z, along with linear terms for each variable and a constant term. It is a more general representation than the center-radius form and requires completing the square to be converted into the form that reveals the sphere's center and radius. The script discusses how to identify and manipulate this form to find the sphere's properties.

💡Quadratic Surface

A quadratic surface is a broader category of geometric shapes in three-dimensional space that includes spheres, ellipsoids, hyperboloids of one and two sheets, and paraboloids. In the video, it is mentioned that if the coefficients of the squared terms in a sphere equation are not all the same, the resulting shape might be a quadratic surface rather than a sphere, indicating a different geometric form.

💡Coefficient

In the context of the video, a coefficient is a numerical factor multiplied by a variable in an algebraic expression. For a sphere equation to represent a sphere, the coefficients of the squared terms (x^2, y^2, z^2) must be the same. The script discusses adjusting coefficients by dividing through by a common factor to standardize the equation and reveal the sphere's properties.

💡Linear Terms

Linear terms in an equation are terms that involve the first power of the variable, such as 2x, 4y, and -6z in the sphere equation. In the video, linear terms are part of the general form of the sphere equation and are addressed during the process of completing the square to help find the sphere's center.

💡Constant Term

A constant term in an algebraic equation is a term that does not contain any variables, such as the -2 in the equation x^2 + y^2 + z^2 + 2x + 4y - 6z - 2 = 0. In the video, the constant term is moved to the other side of the equation during the process of completing the square and contributes to determining the sphere's radius.

💡Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be factored into the square of a binomial. In the video, the goal of completing the square is to transform the squared terms along with their linear terms into perfect square trinomials, which then allows for the sphere equation to be expressed in the center-radius form.

Highlights

Introduction to the concept of sphere equations in calculus, specifically focusing on completing the square.

Explanation of the center-radius form of a sphere equation and its components: center coordinates and radius.

Discussion on the general form of a sphere equation, which may include linear terms and a constant.

The process of transforming a general form sphere equation into the center-radius form using completing the square.

Technique of completing the square by ensuring an expression factors into the same thing twice.

Method to complete the square by taking half of the coefficient of the linear term, squaring it, and adding it to the equation.

Example of completing the square for an expression involving 'x' with a linear term of 10x.

Completing the square for 'y' and 'z' terms with different coefficients, including handling fractions.

Balancing the equation by adding the same value to both sides when completing the square.

Example of finding the center and radius of a sphere given in general form, using the method of completing the square.

Step-by-step walkthrough of completing the square for a sphere equation with x, y, and z terms, and a constant.

Explanation of how to factor trinomials after completing the square to find the center-radius form.

Demonstration of how to identify the center coordinates from the completed square form of the sphere equation.

Calculation of the sphere's radius from the completed square form, including handling of square roots.

Handling of sphere equations with coefficients other than one, by dividing through by the common coefficient.

Final example illustrating the process of finding the center and radius of a sphere with non-unity coefficients.

Conclusion summarizing the importance of completing the square for understanding and solving sphere equations.

Transcripts

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Takeaways

Q & A

What is the standard center-radius form of a sphere equation?

What is the general form of a sphere equation and how does it differ from the center-radius form?

What is the process of completing the square and why is it used in sphere equations?

How do you complete the square for a term like \(x^2 + 10x\)?

What is the center of the sphere given by the equation \(x^2 + y^2 + z^2 + 2x + 4y - 6z - 2 = 0\)?

What is the radius of the sphere with the equation \(x^2 + y^2 + z^2 + 2x + 4y - 6z - 2 = 0\)?

How do you handle coefficients that are not 1 in front of the squared terms when completing the square for a sphere equation?

What is the center of the sphere given by the equation \(3x^2 + 3y^2 + 3z^2 + 6x + y = 0\)?

What is the radius of the sphere with the equation \(3x^2 + 3y^2 + 3z^2 + 6x + y = 0\)?

Why is it necessary to have the same coefficient for all squared terms in a sphere equation?

How do you rationalize the radius expression for a sphere with a fractional radius?