Not using the difference of squares formula correctly

Oct 13, 2025 by windailys.com 57 views

Introduction

Factorizing algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. This technique is essential in solving equations, simplifying expressions, and understanding the properties of functions. In this article, we will focus on factorizing three given expressions: $1.13y^2 + y$ , $1.2x^2 - 10x - 24$ , and $1.39x^2 - y^2 + 10y - 25$ . We will use various factorization techniques, including the greatest common factor (GCF), difference of squares, and grouping.

Factorizing the First Expression: $1.13y^2 + y$

The first expression is a quadratic expression in the variable $y$ . To factorize this expression, we need to find the greatest common factor (GCF) of the two terms. The GCF of $1.13y^2$ and $y$ is $y$ . Therefore, we can factor out $y$ from both terms:

1.13y^2 + y = y(1.13y + 1)

This is the factored form of the first expression.

Factorizing the Second Expression: $1.2x^2 - 10x - 24$

The second expression is also a quadratic expression in the variable $x$ . To factorize this expression, we need to find two numbers whose product is $1.2(-24) = -28.8$ and whose sum is $-10$ . These numbers are $-14.4$ and $2.4$ . Therefore, we can write the expression as:

1.2x^2 - 10x - 24 = 1.2x^2 - 14.4x + 2.4x - 24

Now, we can factor out the GCF of the first two terms, which is $1.2x$ , and the GCF of the last two terms, which is $2.4$ :

1.2x^2 - 10x - 24 = 1.2x(x - 12) + 2.4(x - 12)

Finally, we can factor out the common binomial factor $(x - 12)$ :

1.2x^2 - 10x - 24 = (1.2x + 2.4)(x - 12)

This is the factored form of the second expression.

Factorizing the Third Expression: $1.39x^2 - y^2 + 10y - 25$

The third expression is a quadratic expression in the variables $x$ and $y$ . To factorize this expression, we need to find two numbers whose product is $1.39(-25) = -34.75$ and whose sum is $-y^2 + 10y$ . These numbers are $-y^2 + 10y$ and $-34.75$ . However, we cannot factorize the expression in this way because the two numbers do not have a common factor. Therefore, we need to use the difference of squares formula:

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

In this case, we can write the expression as:

1.39x^2 - y^2 + 10y - 25 = 1.39x^2 - (y^2 - 10y + 25)

Now, we can factor out the difference of squares:

1.39x^2 - y^2 + 10y - 25 = (1.39x^2 - (y - 5)^2)

However, this is not the factored form of the expression. We need to factor out the GCF of the two terms, which is $1.39x^2 - (y - 5)^2$ . Unfortunately, this expression cannot be factored further using the techniques we have learned.

Conclusion

In this article, we have factorized three algebraic expressions using various techniques, including the greatest common factor (GCF), difference of squares, and grouping. We have seen that factorizing expressions is an essential skill in mathematics that can help us solve equations, simplify expressions, and understand the properties of functions. By mastering these techniques, we can tackle a wide range of mathematical problems and challenges.

Glossary of Terms

Greatest Common Factor (GCF): The largest expression that divides two or more expressions without leaving a remainder.
Difference of Squares: A formula that states $a^2 - b^2 = (a + b)(a - b)$ .
Grouping: A technique used to factorize expressions by grouping terms that have common factors.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Q&A: Factorizing Algebraic Expressions

Q: What is factorizing an algebraic expression?

A: Factorizing an algebraic expression involves expressing the expression as a product of simpler expressions. This technique is essential in solving equations, simplifying expressions, and understanding the properties of functions.

Q: What are the different types of factorization techniques?

A: There are several types of factorization techniques, including:

Greatest Common Factor (GCF): This involves finding the largest expression that divides two or more expressions without leaving a remainder.
Difference of Squares: This involves using the formula $a^2 - b^2 = (a + b)(a - b)$ to factorize expressions.
Grouping: This involves grouping terms that have common factors to factorize expressions.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. These numbers are called the roots of the quadratic equation.

Q: What is the difference of squares formula?

A: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$ . This formula can be used to factorize expressions that have a difference of squares.

Q: How do I factorize an expression using the difference of squares formula?

A: To factorize an expression using the difference of squares formula, you need to identify the difference of squares and then use the formula to factorize the expression.

Q: What is the greatest common factor (GCF) of two expressions?

A: The greatest common factor (GCF) of two expressions is the largest expression that divides both expressions without leaving a remainder.

Q: How do I find the GCF of two expressions?

A: To find the GCF of two expressions, you need to list the factors of each expression and then find the largest factor that is common to both expressions.

Q: What is the importance of factorizing algebraic expressions?

A: Factorizing algebraic expressions is essential in solving equations, simplifying expressions, and understanding the properties of functions. It is also used in many real-world applications, such as physics, engineering, and computer science.

Q: Can all algebraic expressions be factorized?

A: No, not all algebraic expressions can be factorized. Some expressions may not have a greatest common factor or may not be able to be factored using the difference of squares formula.

Q: How do I determine if an expression can be factorized?

A: To determine if an expression can be factorized, you need to examine the expression and look for common factors or differences of squares.

Q: What are some common mistakes to avoid when factorizing algebraic expressions?

A: Some common mistakes to avoid when factorizing algebraic expressions include:

Not identifying the greatest common factor (GCF)
Not using the difference of squares formula correctly
Not grouping terms correctly
Not checking for common factors

Conclusion

In this article, we have provided a comprehensive guide to factorizing algebraic expressions, including the greatest common factor (GCF), difference of squares, and grouping. We have also answered some common questions about factorizing algebraic expressions and provided some tips and tricks for avoiding common mistakes. By mastering these techniques, you can tackle a wide range of mathematical problems and challenges.

Glossary of Terms

Greatest Common Factor (GCF): The largest expression that divides two or more expressions without leaving a remainder.
Difference of Squares: A formula that states $a^2 - b^2 = (a + b)(a - b)$ .
Grouping: A technique used to factorize expressions by grouping terms that have common factors.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton