Problem 21 - Technical Mathematics with Calculus

Question

Rewrite standard equation in general form. $$(x+6)^{2}=4(y-5)$$

Step-by-Step Solution

Answer

The general form of the equation is \(x^2 + 12x - 4y + 56 = 0\).

Step 1: Expand the Square

First, expand the square on the left side of the equation. The square of a binomial \(x+6\)^2 is expanded as \(x+6\)\cdot \(x+6\) = \(x^2 + 12x + 36\). The equation becomes \(x^2 + 12x + 36 = 4(y-5)\).

Step 2: Distribute the Constant on the Right

Next, distribute the constant 4 on the right side of the equation across \(y-5\). This gives us \(x^2 + 12x + 36 = 4y - 20\).

Step 3: Move All Terms to One Side

Now, the aim is to have all terms on one side to get the equation in general form \(Ax + By + C = 0\). Subtract \(4y\) and add 20 to both sides so that all the terms are on the left side. The equation becomes \(x^2 + 12x + 36 - 4y + 20 = 0\).

Step 4: Combine Like Terms

Combine like terms, if necessary. Since there are no like terms, the equation, after moving all terms to one side, is the general form: \(x^2 + 12x - 4y + 56 = 0\).

Key Concepts

Expand the Square

The process of expanding the square involves expressing a squared binomial, such as \( (x+6)^2 \), as a trinomial. This step is essential for rewriting equations and solving quadratic problems. To expand \( (x+6)^2 \), we multiply the binomial by itself: \(x+6\) times \(x+6\). This results in the terms \(x^2\), \(6x\), \(6x\) (because each term in the first parenthesis multiplies each term in the second parenthesis), and finally \(36\), which represents \(6\) times \(6\).

After combining these terms, the expanded form is \(x^2 + 12x + 36\). Understanding this process is crucial as it lays the foundation for simplifying and rearranging equations, a skill applicable to a wide range of mathematical topics.

Distributing Constants

The next core concept is distributing constants. In algebra, this refers to the process of multiplying a constant to each term within a parentheses. Let's take the constant 4 in \(4(y-5)\) as an example. Distributing involves multiplying 4 by \(y\) and then by \( -5\), resulting in \(4y\) and \( -20\), respectively.

It is vital to apply this operation carefully to avoid errors that can adversely affect the subsequent steps in solving an equation. This step demonstrates how distributing constants helps transform equations into a more workable form—one of many techniques that enhance a student's aptitude for algebraic manipulation.

Rearranging Algebraic Equations

With both the square expanded and constants distributed, our next task is rearranging algebraic equations. The goal here is to form an equation that sets everything equal to zero—a format known as the general form of a quadratic equation.

In our example, to achieve the standard general form \(Ax^2 + Bx + Cy + D = 0\), we move all terms from one side of the equation to the other. We do this by adding or subtracting terms on both sides, ensuring that the equation remains balanced. By transferring \(4y\) to the left and adding 20 to both sides, we isolate all the variable terms on the left and set them equal to zero. This restructuring is a critical skill in solving more complex equations.

Combining Like Terms

Our final step involves combining like terms, which is the process of simplifying an equation by adding or subtracting terms with the same variable raised to the same power. For instance, if we had terms such as \(12x\) and \(3x\), they could be combined into \(15x\) because they both contain the variable \(x\) to the first power.

In the provided problem, there were no like terms after expanding the square and distributing constants. Hence, the equation \(x^2 + 12x - 4y + 56 = 0\) remains unchanged after this stage. Nevertheless, combining like terms is a fundamental aspect of algebra that helps in simplifying expressions and solving for variables. Remember, the ultimate objective is to streamline the equation, making it easier to work with or solve.