Factoring the Difference of Squares

Oct 13, 2025 by magicalceylontours.com 82 views

Introduction

In algebra, factoring is a crucial concept that helps us simplify complex expressions and solve equations. One of the most important techniques in factoring is the difference of squares, which states that a binomial squared is equal to the square of the first term minus the square of the second term. In this article, we will explore the difference of squares and perfect square trinomials, and learn how to factor each of the given expressions completely over the integers.

What is the Difference of Squares?

The difference of squares is a fundamental concept in algebra that states that:

a^2 - b^2 = (a + b)(a - b)

This formula can be used to factor any expression of the form a^2 - b^2, where a and b are any real numbers.

Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It has the form:

a^2 + 2ab + b^2 = (a + b)^2

a^2 - 2ab + b^2 = (a - b)^2

Factoring the Difference of Squares

Now that we have a good understanding of the difference of squares and perfect square trinomials, let's move on to factoring the given expressions.

1. Factoring 75s + 12

To factor the expression 75s + 12, we need to find two numbers whose product is 75 * 12 and whose sum is 75. These numbers are 60 and 15, since 60 * 15 = 900 and 60 + 15 = 75.

75s + 12 = 60s + 15s + 12 = 3(20s + 4) + 3(5s + 4) = 3(20s + 5s + 4) = 3(25s + 4)

Therefore, the factored form of 75s + 12 is 3(25s + 4).

2. Factoring 75s^2 + 12

To factor the expression 75s^2 + 12, we need to find two numbers whose product is 75 * 12 and whose sum is 75s^2. These numbers are 60s^2 and 15s^2, since 60s^2 * 15s^2 = 900s^4 and 60s^2 + 15s^2 = 75s^2.

75s^2 + 12 = 60s^2 + 15s^2 + 12 = 3(20s^2 + 5s^2 + 4) = 3(25s^2 + 4)

Therefore, the factored form of 75s^2 + 12 is 3(25s^2 + 4).

3. Factoring 75s^2 + 12

75s^2 + 12 = 60s^2 + 15s^2 + 12 = 3(20s^2 + 5s^2 + 4) = 3(25s^2 + 4)

Therefore, the factored form of 75s^2 + 12 is 3(25s^2 + 4).

Conclusion

In this article, we have learned about the difference of squares and perfect square trinomials, and how to factor each of the given expressions completely over the integers. We have seen how to use the difference of squares formula to factor expressions of the form a^2 - b^2, and how to factor perfect square trinomials into the square of a binomial. By mastering these techniques, we can simplify complex expressions and solve equations with ease.

Key Takeaways

The difference of squares formula states that a^2 - b^2 = (a + b)(a - b).
A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial.
To factor an expression, we need to find two numbers whose product is the product of the coefficients and whose sum is the sum of the coefficients.
The factored form of an expression is the product of the two numbers found in the previous step.

Practice Problems

Factor the expression 16x^2 + 24x + 9.
Factor the expression 9x^2 - 12x + 4.
Factor the expression 25x^2 + 30x + 9.

Solutions

16x^2 + 24x + 9 = (4x + 3)^2
9x^2 - 12x + 4 = (3x - 2)^2
25x^2 + 30x + 9 = (5x + 3)^2

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for general algebra and calculus textbooks, and are not specific to the topic of difference of squares and perfect square trinomials.