Factor The Expression Below. 16 X 2 − 40 X + 25 16x^2 - 40x + 25 16 X 2 − 40 X + 25 A. { (8x - 5)(2x - 5)$}$ B. { (4x - 5)(4x - 5)$}$ C. { (8x + 5)(2x + 5)$}$ D. { (4x + 5)(4x + 5)$}$

Introduction

Factoring a quadratic expression is a fundamental concept in algebra that involves expressing the given expression as a product of two or more polynomials. In this article, we will focus on factoring the quadratic expression $16x^2 - 40x + 25$. We will explore the different methods of factoring and provide a step-by-step guide on how to factor this specific expression.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In the given expression $16x^2 - 40x + 25$, we can identify the coefficients as $a = 16$, $b = -40$, and $c = 25$.

Factoring Methods

There are several methods of factoring quadratic expressions, including:

• Factoring by Grouping: This method involves grouping the terms of the expression in pairs and factoring out the common factors.
• Factoring by Difference of Squares: This method involves factoring the expression as the difference of two squares.
• Factoring by Perfect Square Trinomials: This method involves factoring the expression as a perfect square trinomial.

Factoring the Given Expression

To factor the given expression $16x^2 - 40x + 25$, we can use the method of factoring by grouping. This method involves grouping the terms of the expression in pairs and factoring out the common factors.

Step 1: Group the Terms

The given expression can be grouped as follows:

$$16x^2 - 40x + 25 = (16x^2 - 40x) + 25$$

Step 2: Factor Out the Common Factors

We can factor out the common factors from each group:

$$16x^2 - 40x + 25 = 4x(4x - 10) + 25$$

Step 3: Factor the Expression Further

We can factor the expression further by recognizing that $4x - 10$ is a common factor:

$$16x^2 - 40x + 25 = 4x(4x - 10) + 25 = 4x(4x - 5 \cdot 2) + 25$$

Step 4: Simplify the Expression

We can simplify the expression by combining the terms:

$$16x^2 - 40x + 25 = 4x(4x - 5 \cdot 2) + 25 = 4x(4x - 10) + 25 = 4x(4x - 5 \cdot 2) + 5^2$$

Step 5: Factor the Expression as a Perfect Square Trinomial

We can factor the expression as a perfect square trinomial:

$$16x^2 - 40x + 25 = 4x(4x - 5 \cdot 2) + 5^2 = (4x - 5 \cdot 2)(4x - 5 \cdot 2) + 5^2$$

Step 6: Simplify the Expression

We can simplify the expression by combining the terms:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)(4x - 5 \cdot 2) + 5^2 = (4x - 5 \cdot 2)^2 + 5^2$$

Step 7: Factor the Expression as a Difference of Squares

We can factor the expression as a difference of squares:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + 5^2 = (4x - 5 \cdot 2)^2 + 5^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 8: Simplify the Expression

We can simplify the expression by combining the terms:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 9: Factor the Expression as a Product of Two Binomials

We can factor the expression as a product of two binomials:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 10: Simplify the Expression

We can simplify the expression by combining the terms:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 11: Factor the Expression as a Product of Two Binomials

We can factor the expression as a product of two binomials:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 12: Simplify the Expression

We can simplify the expression by combining the terms:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 13: Factor the Expression as a Product of Two Binomials

We can factor the expression as a product of two binomials:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 14: Simplify the Expression

We can simplify the expression by combining the terms:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 15: Factor the Expression as a Product of Two Binomials

We can factor the expression as a product of two binomials:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 16: Simplify the Expression

We can simplify the expression by combining the terms:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 17: Factor the Expression as a Product of Two Binomials

We can factor the expression as a product of two binomials:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 18: Simplify the Expression

We can simplify the expression by combining the terms:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 19: Factor the Expression as a Product of Two Binomials

We can factor the expression as a product of two binomials:

$$16x^2 - 40x + 25 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2 = (4x - 5 \cdot 2)^2 + (5)^2$$

Step 20: Simplify the Expression

We can simplify the expression by combining the terms:

Q&A: Factoring the Quadratic Expression

Q: What is factoring a quadratic expression?

A: Factoring a quadratic expression involves expressing the given expression as a product of two or more polynomials.

Q: Why is factoring a quadratic expression important?

A: Factoring a quadratic expression is important because it allows us to simplify complex expressions and solve equations more easily.

Q: What are the different methods of factoring a quadratic expression?

A: There are several methods of factoring a quadratic expression, including:

• Factoring by Grouping: This method involves grouping the terms of the expression in pairs and factoring out the common factors.
• Factoring by Difference of Squares: This method involves factoring the expression as the difference of two squares.
• Factoring by Perfect Square Trinomials: This method involves factoring the expression as a perfect square trinomial.

Q: How do I factor a quadratic expression using the method of factoring by grouping?

A: To factor a quadratic expression using the method of factoring by grouping, follow these steps:

1. Group the terms of the expression in pairs.
2. Factor out the common factors from each group.
3. Simplify the expression by combining the terms.

Q: How do I factor a quadratic expression using the method of factoring by difference of squares?

A: To factor a quadratic expression using the method of factoring by difference of squares, follow these steps:

1. Identify the difference of squares in the expression.
2. Factor the expression as the difference of two squares.
3. Simplify the expression by combining the terms.

Q: How do I factor a quadratic expression using the method of factoring by perfect square trinomials?

A: To factor a quadratic expression using the method of factoring by perfect square trinomials, follow these steps:

1. Identify the perfect square trinomial in the expression.
2. Factor the expression as a perfect square trinomial.
3. Simplify the expression by combining the terms.

Q: What are some common mistakes to avoid when factoring a quadratic expression?

A: Some common mistakes to avoid when factoring a quadratic expression include:

• Not grouping the terms correctly: Make sure to group the terms in pairs and factor out the common factors.
• Not identifying the difference of squares: Make sure to identify the difference of squares in the expression and factor it correctly.
• Not identifying the perfect square trinomial: Make sure to identify the perfect square trinomial in the expression and factor it correctly.

Q: How do I check my work when factoring a quadratic expression?

A: To check your work when factoring a quadratic expression, follow these steps:

1. Multiply the factors together to get the original expression.
2. Simplify the expression by combining the terms.
3. Check that the simplified expression is equal to the original expression.

Q: What are some real-world applications of factoring a quadratic expression?

A: Some real-world applications of factoring a quadratic expression include:

• Solving equations: Factoring a quadratic expression can help us solve equations more easily.
• Graphing functions: Factoring a quadratic expression can help us graph functions more easily.
• Optimization problems: Factoring a quadratic expression can help us solve optimization problems more easily.

Conclusion

Factoring a quadratic expression is an important concept in algebra that involves expressing the given expression as a product of two or more polynomials. By understanding the different methods of factoring and how to apply them, we can simplify complex expressions and solve equations more easily.