Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, determine which conic section is represented based on the given equation. \(x^{2}-10 x+4 y-10=0\)

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Rearrange the Equation

Start by rearranging the given equation to identify standard terms. The equation is given as \[ x^2 - 10x + 4y - 10 = 0. \] Firstly, let's isolate the conic term by adding 10 to both sides: \[ x^2 - 10x + 4y = 10. \]

Step 2: Complete the Square for x

To identify the conic section, complete the square for the \(x\) terms. Consider the \(x\) terms: \[ x^2 - 10x. \] The coefficient of \(x\) is \(-10\), so half of \(-10\) is \(-5\), and \((-5)^2 = 25\). Add and subtract 25 inside the equation, keeping it balanced: \[ (x^2 - 10x + 25) - 25 + 4y = 10. \] This can be rewritten as: \[ (x - 5)^2 + 4y = 35. \]

Step 3: Solve the Equation to Identify the Conic

Rearrange the equation to highlight any transformations of the known conic forms. From Step 2, we have: \[ (x - 5)^2 + 4y = 35. \] Subtract \(4y\) on both sides to express \(y\): \[ 4y = 35 - (x - 5)^2. \] Divide everything by 4 to isolate \(y\) and see a clearer conic section: \[ y = \frac{35 - (x - 5)^2}{4}. \]

Step 4: Identify the Conic Section

The rearranged equation \( y = \frac{35 - (x - 5)^2}{4} \) signifies a vertical parabola because it can be interpreted as \( (x - h)^2 = 4p(y - k) \) upon comparing with the standard form of a parabola. The equation clearly isolates \(y\) in terms of \(x\), which is a characteristic of a parabola.

Key Concepts

Parabola

A parabola is a U-shaped curve that is defined as a collection of points equidistant from a fixed point called the "focus" and a line called the "directrix." Parabolas can open upwards, downwards, leftwards, or rightwards, depending on their orientation. In algebraic form, a parabola is often represented by equations like

• Standard form: \( y = ax^2 + bx + c \)
• Vertex form: \( y = a(x - h)^2 + k \)

This particular problem resulted in finding a vertical parabola. When you worked through the rearranged equation, \[ y = \frac{35 - (x - 5)^2}{4} \] it followed the form \( (x - h)^2 = 4p(y - k) \), where

• \( h = 5 \),
• \( k \) = 0 (after simplification),
• \( p \) is determined by the 4 in the denominator, indicating how "wide" or "narrow" the parabola is.

Understanding these parameters helps identify the parabola's orientation and position on the graph.

Complete the Square

Completing the square is a method used in algebra to transform a quadratic expression into a perfect square trinomial. This technique makes it easier to solve, graph, or identify the properties of quadratic equations. The process involves three main steps:

• Identify the coefficient of the linear term, in this case, \( x \).
• Take half of this coefficient, and square it. This value gets added and subtracted within the equation to form a complete square.
• Re-write the equation in perfect square form.

In our exercise, completing the square transformed \( x^2 - 10x \) into\( (x - 5)^2 - 25 \).This step is crucial because it helps rearrange the equation into a form that easily leads to identification of conic sections such as the parabola. By rewriting the quadratic portion as a squared binomial, one can simplify solving or graphing the expression.

Algebraic Equations

Algebraic equations form the foundation for solving problems involving unknowns. They consist of expressions set equal to each other and can involve constants, variables, and arithmetic operations. There are different types of algebraic equations based on their degree and terms.

• A linear equation has the form \( ax + by + c = 0 \).
• Quadratic equations are expressed as \( ax^2 + bx + c = 0 \).
• Higher-order polynomials involve terms like \( x^3 \) or larger exponents.

In conic section analysis, algebraic equations are used to determine the type of conic based on their structure. In our exercise, recognizing the structure of the quadratic form of \( x^2 - 10x + 4y - 10 = 0 \)was essential. By manipulating this algebraic equation through steps like completing the square, identifying the conic section becomes straightforward. Understanding these transformations is key for dealing with various algebraic challenges in conics and beyond.