Problem 29 - Algebra A Combined Function

Question

Factor each completely. \(x^{2}-25\)

Step-by-Step Solution

Answer

The expression factors to \((x - 5)(x + 5)\).

Step 1: Identify the Type of Expression

The expression is a difference of squares, which has the form of \(a^2 - b^2\), where \(a = x\) and \(b = 5\).

Step 2: Apply the Difference of Squares Formula

The difference of squares can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).

Step 3: Substitute and Factor

Substitute \(a = x\) and \(b = 5\) into the formula: \(x^2 - 5^2 = (x - 5)(x + 5)\).

Key Concepts

Difference of Squares

In algebra, a powerful factoring technique is the difference of squares. It's a special type of expression that takes the form of \(a^2 - b^2\). What makes this expression so unique is that it can be neatly factored into a product of two binomials: \((a - b)(a + b)\).
To recognize a difference of squares in a math problem:

• Identify if both terms are perfect squares.
• Look for a subtraction operation between them.

When you break down the expression \(x^2 - 25\), you see that \(x\) is the square root of \(x^2\) and 5 is the square root of 25. This is a textbook example of a difference of squares. Once identified, it can be factored into \((x - 5)(x + 5)\).
This efficient method not only simplifies the expression but also is a handy tool in solving quadratic equations. Understanding and mastering the difference of squares can make various algebra problems much easier to tackle.

Algebra

Algebra is like the language of mathematics. It uses symbols and letters to represent numbers and quantities in formulas and equations. The purpose is to find the unknowns and understand patterns.
In algebra, different types of expressions and equations are tackled with strategic methods. The given exercise \(x^2 - 25\) is a polynomial expression, a type common in algebra. Here, identifying patterns, like the difference of squares, allows straightforward factoring into simpler expressions.
Algebra isn't just about numbers; it's about thinking logically and solving problems with rules and methods that transcend basic arithmetic. By mastering algebra, students unlock the ability to handle complex math problems and gain a deeper understanding of how math influences the world around them.

Factoring Techniques

Factoring techniques are like tools in your math toolkit that allow you to break down complex expressions into simpler parts. Factoring is essential for solving equations, simplifying expressions, and understanding mathematical structures.
There are several factoring techniques, each suited for different types of expressions:

• Greatest Common Factor (GCF): Factor out the largest common factor shared by all terms.
• Trinomials: Use specific patterns or the quadratic formula to factor.
• Difference of Squares: As shown in \(x^2 - 25 = (x - 5)(x + 5)\).
• Completing the Square: Manipulate a quadratic equation into a perfect square trinomial.

Factoring not only makes equations easier to solve, but it also enhances understanding of the relationships between components of an expression. By practicing different factoring techniques, students acquire flexible strategies to approach and solve various mathematical challenges.