Factor Completely.${ 4y^2 - 20y + 25 }$

Understanding Quadratic Expressions

Quadratic expressions are a fundamental concept in algebra, and factoring them is a crucial skill to master. A quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. In this article, we will focus on factoring the quadratic expression $4y^2 - 20y + 25$. Factoring quadratic expressions involves expressing them as a product of two binomials, which can be a challenging task, but with the right techniques and strategies, it can be achieved.

The Importance of Factoring Quadratic Expressions

Factoring quadratic expressions is essential in algebra because it allows us to solve equations and inequalities, find the roots of a quadratic equation, and simplify complex expressions. When we factor a quadratic expression, we can identify its roots, which are the values of the variable that make the expression equal to zero. This is particularly useful in solving quadratic equations, where the roots of the equation are the solutions to the equation.

The Process of Factoring Quadratic Expressions

To factor a quadratic expression, we need to follow a series of steps. The first step is to look for any common factors in the expression. If there are any common factors, we can factor them out of the expression. The next step is to look for two binomials whose product is equal to the original expression. We can use the FOIL method to check if two binomials are factors of the expression. The FOIL method involves multiplying the first terms of the two binomials, then multiplying the outer terms, then multiplying the inner terms, and finally multiplying the last terms.

Factoring the Quadratic Expression $4y^2 - 20y + 25$

Now that we have a good understanding of the process of factoring quadratic expressions, let's apply it to the expression $4y^2 - 20y + 25$. The first step is to look for any common factors in the expression. In this case, there are no common factors, so we can move on to the next step. The next step is to look for two binomials whose product is equal to the original expression. We can use the FOIL method to check if two binomials are factors of the expression.

Using the FOIL Method to Factor the Quadratic Expression

To use the FOIL method, we need to multiply the first terms of the two binomials, then multiply the outer terms, then multiply the inner terms, and finally multiply the last terms. Let's assume that the two binomials are $(2y + a)$ and $(2y + b)$. We can multiply the first terms of the two binomials, which gives us $2y \cdot 2y = 4y^2$. We can multiply the outer terms of the two binomials, which gives us $2y \cdot b = 2yb$. We can multiply the inner terms of the two binomials, which gives us $a \cdot 2y = 2ay$. Finally, we can multiply the last terms of the two binomials, which gives us $a \cdot b = ab$.

Simplifying the Expression Using the FOIL Method

Now that we have multiplied the terms of the two binomials, we can simplify the expression. We can combine like terms, which gives us $4y^2 + 2yb + 2ay + ab$. We can see that the expression contains the terms $4y^2$, $2yb$, $2ay$, and $ab$. We can combine the like terms, which gives us $4y^2 + 2y(b + a) + ab$.

Factoring the Quadratic Expression

Now that we have simplified the expression, we can factor it. We can see that the expression contains the terms $4y^2$, $2y(b + a)$, and $ab$. We can factor out the greatest common factor of the terms, which is $2y$. We can factor out the $2y$ from the first two terms, which gives us $2y(2y + b + a)$. We can see that the expression contains the term $ab$, which is the product of the two binomials. We can factor the expression as $(2y + a)(2y + b)$.

Conclusion

In this article, we have factored the quadratic expression $4y^2 - 20y + 25$. We have used the FOIL method to check if two binomials are factors of the expression. We have simplified the expression using the FOIL method and factored it. The final factored form of the expression is $(2y - 5)^2$. Factoring quadratic expressions is an essential skill in algebra, and with the right techniques and strategies, it can be achieved.

Final Answer

The final answer is: $\boxed{(2y - 5)^2}$

Understanding Quadratic Expressions and Factoring

Quadratic expressions are a fundamental concept in algebra, and factoring them is a crucial skill to master. A quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. In this article, we will focus on factoring the quadratic expression $4y^2 - 20y + 25$. Factoring quadratic expressions involves expressing them as a product of two binomials, which can be a challenging task, but with the right techniques and strategies, it can be achieved.

Q&A: Factoring Quadratic Expressions

Q: What is the difference between factoring and simplifying a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while simplifying a quadratic expression involves combining like terms to reduce the expression to its simplest form.

Q: How do I determine if a quadratic expression can be factored?

A: To determine if a quadratic expression can be factored, you need to look for any common factors in the expression. If there are any common factors, you can factor them out of the expression. You can also use the FOIL method to check if two binomials are factors of the expression.

Q: What is the FOIL method?

A: The FOIL method is a technique used to check if two binomials are factors of a quadratic expression. It involves multiplying the first terms of the two binomials, then multiplying the outer terms, then multiplying the inner terms, and finally multiplying the last terms.

Q: How do I use the FOIL method to factor a quadratic expression?

A: To use the FOIL method, you need to multiply the first terms of the two binomials, then multiply the outer terms, then multiply the inner terms, and finally multiply the last terms. You can then simplify the expression by combining like terms.

Q: What is the difference between factoring a quadratic expression and factoring a polynomial?

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while factoring a polynomial involves expressing it as a product of two or more polynomials.

Q: How do I factor a quadratic expression with a negative coefficient?

A: To factor a quadratic expression with a negative coefficient, you need to use the same techniques as factoring a quadratic expression with a positive coefficient. However, you need to be careful when multiplying the terms to ensure that the signs are correct.

Q: Can a quadratic expression be factored if it has no common factors?

A: Yes, a quadratic expression can be factored even if it has no common factors. You can use the FOIL method to check if two binomials are factors of the expression.

Conclusion

In this article, we have provided a step-by-step guide to factoring quadratic expressions, including a Q&A section to address common questions and concerns. Factoring quadratic expressions is an essential skill in algebra, and with the right techniques and strategies, it can be achieved.

Final Answer

The final answer is: $\boxed{(2y - 5)^2}$