Problem 50 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ x^{2}-25=0 $$

Step-by-Step Solution

Answer

Answer: The given equation is of degree 2 and is classified as a quadratic equation.

Step 1: Write down the given equation

The given equation is $$ x^2 - 25 = 0 $$

Step 2: Identify the highest power

The highest power of the variable (x) in the given equation is 2, coming from the term \(x^2\). Thus, the degree of this equation is equal to its highest power, which is 2.

Step 3: Classify according to the degree

Since the degree of the equation is 2, we can classify it as a quadratic equation. Quadratic equations have the general form of \(ax^2 + bx + c = 0\), where a, b, and c are constants, and a is not equal to 0. Our equation can be written as $$ 1x^2 + 0x - 25 = 0 $$

Step 4: Conclusion

The given equation \(x^2 - 25 = 0\) is a quadratic equation of degree 2.

Key Concepts

Understanding Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. These symbols can represent numbers, shapes, and operations, allowing us to form equations and solve problems. It provides a systematic way to express and solve mathematical computations. In algebra, variables stand for unknown values and can be used to formulate equations and expressions.

• **Variables**: These are symbols, often letters, used to represent unknown values or quantities that can change.
• **Constants**: Numbers or fixed values that do not change.
• **Coefficients**: Numbers that multiply the variables.

Through algebra, we're able to explore relationships between variables and constants in various forms of equations, whether they're simple linear equations or more complex polynomial equations.

Degree of Equations

The degree of an equation is an important concept that helps us understand the nature of the polynomial. It is determined by the highest power (exponent) of the variable present in the equation. Knowing the degree of an equation is key in polynomial classification and solving the equation effectively.

• **Linear Equations**: These have the highest power of 1, for example, the equation \(x + 3 = 0\) is linear.
• **Quadratic Equations**: These have the highest power of 2, as seen in the equation \(x^2 - 25 = 0\), meaning the solution can produce up to two roots or solutions.
• **Cubic Equations**: These have the highest power of 3, meaning they can have up to three roots.

Once you identify the degree of the equation, you can predict how many solutions the equation might have and what methods are best suited to solve it.

Polynomial Classification

Polynomial classification involves organizing polynomials based on their degree, which leads to easier identification and solving of these equations. Polynomials consist of terms that each contain a constant multiplied by a variable raised to a non-negative integer power.

• **Monomial**: A polynomial with just one term, such as \(5x^3\).
• **Binomial**: A polynomial with two terms, like \(x^2 - 25\). This could also represent expressions such as \((x - 5)(x + 5)\), which when factored, help in solving quadratic equations like the one mentioned.
• **Trinomial**: A polynomial with three terms, such as \(x^2 + 5x + 6\).

Classifying these polynomials helps in deciding which algebraic methods are best suited to rearrange, simplify, and solve them. This step is essential in understanding the nature and structure of equations.