Problem 20 - Elementary Algebra

Question

For the following problems, find the products. $$ (a-10)^{2} $$

Step-by-Step Solution

Answer

Answer: \(a^2 - 20a + 100\)

Step 1: Square the Binomial

To find the product of \((a-10)^2\), use the formula for squaring a binomial, which is \((a-b)^2 = a^2 - 2ab + b^2\). In our case, we have \(a\) and \(b=10\). So, the expression becomes: \((a-10)^2 = a^2 - 2(a)(10) + 10^2\).

Step 2: Simplify the expression

After applying the formula, we need to simplify the expression: $$a^2 - 2(a)(10) + 10^2 = a^2 - 20a + 100$$. So, the product of \((a-10)^2\) is \(a^2 - 20a + 100\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all mathematics. With algebra, we can formulate real-world problems into mathematical expressions and equations to find solutions. Mathematics often use letters instead of numbers, such as in this example with the expression \(a - 10\). Algebraic expressions can consist of constants, variables, and operations such as addition, subtraction, multiplication, and division. When we manipulate these expressions, we use specific rules and properties, like the distributive property when factoring or expanding expressions. Understanding these rules is crucial for solving more complex equations down the road.Algebra is not only about the numbers and symbols but also about the logic and principles behind manipulating them. As students progress, they learn more about structures such as groups, rings, and fields, which further extend the foundational ideas of algebra.

Quadratic Expressions

Quadratic expressions are polynomial expressions of the second degree. This means they have a variable raised to the power of two. The general form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.In our exercise, the result of squaring the binomial \(a - 10\) resulted in the quadratic expression \(a^2 - 20a + 100\). Each term in this expression has a specific role:

• The term \(a^2\) represents the squared variable.
• The term \(-20a\) represents the linear term, derived from multiplying the variable by a constant.
• The constant term \(+100\) stands alone without a variable.

Quadratic expressions are fundamental in algebra since they appear in various types of problems, such as calculating areas, projectile motion, and even financial forecasting. Solving quadratic equations often involves techniques such as factoring, completing the square, or using the quadratic formula.

Factoring

Factoring is an algebraic process in which a polynomial is expressed as a product of its factors. Factors are simpler polynomials or numbers that can be multiplied together to yield the original expression. By factoring expressions, we can solve quadratic equations and simplify complex polynomials. For instance, if we need to factor the quadratic expression \(a^2 - 20a + 100\), we look for two binomials such that their product equals the original expression. Factoring becomes a problem of recognizing patterns and applying algebraic identities, like factoring squares of binomials, difference of squares, or trinomials.Factoring is a key skill in algebra, offering solutions to equations and insights into number theory and relationships between numbers. It also simplifies the calculation process and is a critical step in simplifying mathematical expressions for further analysis.