Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Write the quadratic equation in general form. $$ x^{2}=25 x $$

Step-by-Step Solution

Answer

The quadratic equation in general form is \(x^{2} - 25x + 0 = 0\)

Step 1: Rearrange the terms

To get the equation into the general form, subtract \(25x\) from both sides of the equation. This results in \(x^{2} - 25x = 0\) which corresponds to the form \(ax^{2} + bx = 0\) where \(a = 1\) and \(b = -25\).

Step 2: Add constant term

The general form includes a constant term 'c'. In this case, the constant term is 0 since it isn't given in our equation. We therefore have \(x^{2} - 25x + 0 = 0\). Now, our equation is in the general form \(ax^{2} + bx + c = 0\) with \(a = 1\), \(b = -25\), and \(c = 0\).

Key Concepts

General Form

The general form of a quadratic equation is an expression that allows us to identify the coefficients of the equation easily. It is structured as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

Having a quadratic equation in this form is vital for solving it using various methods, such as factoring, completing the square, or applying the quadratic formula. The coefficients \(a\), \(b\), and \(c\) are key components that help us understand the properties and solutions of the quadratic equation.

For instance, in the equation \(x^2 - 25x = 0\), we rearranged it to the general form \(x^2 - 25x + 0 = 0\), making it clear that \(a = 1\), \(b = -25\), and \(c = 0\). This form makes it straightforward to identify each term's role in the equation's graph and solutions.

Rearranging Terms

Rearranging terms is a crucial step in converting a given equation into its general form. This involves moving all terms to one side of the equation to set them equal to zero.

In the original equation \(x^2 = 25x\), we see that it isn't in the general form because not all of the terms are on one side equal to zero.

By subtracting \(25x\) from both sides, we rearrange the equation as \(x^2 - 25x = 0\). The process of rearranging terms ensures that we have the quadratic terms aligned properly, facilitating further simplification or solution-finding techniques. Effective rearranging is essential for correctly identifying the coefficients \(a\), \(b\), and \(c\) needed in the quadratic formula or other solving methods.

Polynomial Equations

Quadratic equations are a type of polynomial equation that specifically feature the degree of 2, meaning the highest power of the variable (commonly \(x\)) is squared.

Polynomial equations can have degrees from zero to infinity, but quadratics are uniquely significant due to their parabolic shape when graphed and their application in various real-world scenarios.

In our given scenario, the original equation \(x^2 = 25x\) represented a quadratic polynomial equation. The expression \(x^2 - 25x + 0 = 0\) maintains its structure as a polynomial equation, specifically quadratic, since it adheres to the form \(ax^2 + bx + c = 0\). Recognizing such equations and converting them to the necessary form is foundational in algebra and precalculus studies, open the door to graphing and finding solutions easily.

• Definitions and recognition of polynomial degrees can aid in identifying the kind of solutions or graph shape expected from the equation.
• Quadratics are just one stepping stone in polynomial studies but are fundamental to many further math concepts.