Problem 67 - Beginning and Intermediate Algebra

Question

Factor completely. $$4 a^{2}-100$$

Step-by-Step Solution

Answer

The completely factored expression for \(4a^2 - 100\) is \((2a + 10)(2a - 10)\).

Step 1: Identify the difference of squares

The given expression is $$4a^2 - 100$$. Notice that both terms are perfect squares: - \(4a^2\) is \((2a)^2\) - \(100\) is \(10^2\) So, the expression can be written as a difference of squares: $$(2a)^2 - (10)^2$$

Step 2: Apply the difference of squares formula

Using the formula $$a^2-b^2 = (a+b)(a-b)$$, we can factor the given expression as follows: - Let \(a = 2a\) - Let \(b = 10\) Now, apply the formula: $$ (2a + 10)(2a - 10) $$

Step 3: Write the final factored expression

After applying the difference of squares formula, we get the factored expression as: $$ (2a + 10)(2a - 10) $$ So, the completely factored expression for $$4a^2 - 100$$ is $$ (2a + 10)(2a - 10) $$.

Key Concepts

Difference of Squares

The concept of the "difference of squares" is a fundamental idea in algebra that significantly simplifies the process of factoring certain polynomial expressions. At its core, the difference of squares describes an algebraic expression of the form \(a^2 - b^2\). This expression can be factored elegantly using a simple formula:

• \(a^2 - b^2 = (a+b)(a-b)\)

This formula is derived from the idea that when two identical terms are multiplied by their conjugates, the middle terms cancel out, only leaving the squares. To identify if an expression fits this pattern, both terms should be perfect squares. For example, in the expression \(4a^2 - 100\), \(4a^2\) is a perfect square \((2a)^2\) and \(100\) is \(10^2\). Recognizing such patterns can be very useful in algebra problems as it allows for quicker simplification.

Algebra Problems

Solving algebra problems often involves recognizing patterns and applying formulas, as seen in the process of factoring the expression \(4a^2 - 100\). Algebra problems can range from simple equations to more complex polynomial expressions. Factoring is a key skill that supports solving various types of algebra problems. When faced with a problem:

• Identify terms that can be factored using known formulas, like the difference of squares.
• Rewrite the problem into smaller parts to simplify.

In the case of the example, after recognizing \(4a^2 - 100\) as a difference of squares, the expression simplifies quickly. This shows how understanding broader concepts and patterns of algebra can significantly ease the calculation and solution in math problems.

Polynomial Expressions

Polynomial expressions consist of variables and constants combined using addition, subtraction, and multiplication. Understanding how to manipulate these expressions, especially factoring, is crucial not only in algebra but across various branches of mathematics. Factoring polynomials, for instance, involves breaking down the expression into simpler components or terms that multiply to form the original expression.

• One common technique is identifying special products, like perfect squares or cubes, within the polynomial.

In the given expression \(4a^2 - 100\), it is essential to understand that polynomial expressions might contain hidden factors like squares that can be extracted using factorization techniques.This knowledge of polynomial manipulation is important for solving not only algebraic equations but also real-world problems modeled by these equations.