Problem 20 - Intermediate Algebra

Question

\(y^{2}-8 y-x^{2}-4 x+3=0\)

Step-by-Step Solution

Answer

The equation is a hyperbola in the form \((y - 4)^2 - (x + 2)^2 = 13\).

Step 1: Rearrange the Equation

Start by rearranging the given equation to separate terms involving x and y on one side: \[ y^2 - 8y - (x^2 + 4x) = -3 \]

Step 2: Complete the Square for y

To complete the square for the terms in y, we focus on \(y^2 - 8y\). We take half of the coefficient of y, which is 8, divide by 2 to get 4, and square it to get 16. Add and subtract this inside the equation:\[ (y^2 - 8y + 16) - 16 - (x^2 + 4x) = -3 \]

Step 3: Simplify y Terms

The expression \(y^2 - 8y + 16\) can be rewritten as a perfect square: \[ (y - 4)^2 \] This simplifies to: \[ (y - 4)^2 - 16 - (x^2 + 4x) = -3 \]

Step 4: Complete the Square for x

Next, complete the square for the x terms: \(x^2 + 4x\). Take half of 4 (which is 2), and square it to obtain 4. Add and subtract 4:\[ (y - 4)^2 - 16 - (x^2 + 4x + 4 - 4) = -3 \]

Step 5: Simplify x Terms

The expression \(x^2 + 4x + 4\) can be rewritten as a perfect square:\[ (x + 2)^2 \]Substitute back into the equation:\[ (y - 4)^2 - (x + 2)^2 - 16 = -3 \]

Step 6: Simplify the Equation

Combine constants on the right side:\[ (y - 4)^2 - (x + 2)^2 = 13 \] This is the equation of a hyperbola in standard form.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form that reveals more information about it, especially useful in solving problems related to conic sections.
When you have a quadratic equation like \( y^2 - 8y \), you start by taking the coefficient of the linear term \( y \), which is -8.
To complete the square:

• Divide by 2: \( -8/2 = -4 \)
• Square the result: \( (-4)^2 = 16 \)

Add and subtract this square inside your expression.
Doing so lets you express this quadratic in its factored form, as \( (y - 4)^2 \), exposing the root and vertex nature of the quadratic component.
Completing the square essentially simplifies understanding of the quadratic through algebraic manipulation.

Equation of Conic Sections

Conic sections, such as circles, ellipses, parabolas, and hyperbolas, are shapes formed by slicing a cone at different angles.
Each section has its own characteristic equation.
For hyperbolas, the general form is \( (y-k)^2/a^2 - (x-h)^2/b^2 = 1 \).
In the problem, our goal is to convert the original equation into this form by completing the square for both \(x\) and \(y\).
By transforming \( y^2 - 8y \) and \( x^2 + 4x \), we rewrite the equation

• as \( (y - 4)^2 - (x + 2)^2 = 13 \)

Although the constant on the right side is not 1, we recognize this as a form of a hyperbola.
Understanding the form of conic sections enables us to identify their properties, such as vertices and axes of symmetry.

Intermediate Algebra

Intermediate algebra often involves the manipulation of equations and expressions to explore relationships between variables. Understanding these foundations is crucial in topics like conic sections.
It involves applying different strategies like factoring, expanding, and completing the square.
In our solution, several steps use these methods to rearrange and simplify the equation.

• After manipulating terms to form complete squares, the equation is easier to interpret as a standard form of a hyperbola.
• This involves recognizing patterns and applying algebraic identities such as the square of a binomial.

Practicing intermediate algebra enhances problem-solving skills and prepares you for more advanced math topics.
It emphasizes a systematic approach to break down complex problems into manageable steps.