Expand The Following Product: { (2x - 5)(2x + 6)$}$Answer: { \square$}$

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Introduction

In algebra, expanding a product of two binomials is a crucial skill that helps in solving various mathematical problems. The given product, $(2x - 5)(2x + 6)$, is a quadratic expression that can be expanded using the distributive property. In this article, we will learn how to expand the given product and simplify it to its final form.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

This property can be applied to expand a product of two binomials. To expand the given product, we will use the distributive property to multiply each term in the first binomial by each term in the second binomial.

Expanding the Product

To expand the product $(2x - 5)(2x + 6)$, we will multiply each term in the first binomial by each term in the second binomial.

$(2x - 5)(2x + 6) = (2x)(2x) + (2x)(6) - (5)(2x) - (5)(6)$

Simplifying the Expression

Now, we will simplify the expression by combining like terms.

$(2x)(2x) = 4x^2$

$(2x)(6) = 12x$

$-(5)(2x) = -10x$

$-(5)(6) = -30$

Combining Like Terms

Now, we will combine the like terms to simplify the expression.

$4x^2 + 12x - 10x - 30$

Final Form

Combining the like terms, we get:

$4x^2 + 2x - 30$

Conclusion

In this article, we learned how to expand the given product $(2x - 5)(2x + 6)$ using the distributive property. We simplified the expression by combining like terms and arrived at the final form of the expression, which is $4x^2 + 2x - 30$. This is a quadratic expression that can be used to solve various mathematical problems.

Examples and Applications

The expanded form of the given product has various applications in mathematics and real-world problems. Here are a few examples:

• Quadratic Equations: The expanded form of the given product can be used to solve quadratic equations. For example, if we have a quadratic equation in the form of $ax^2 + bx + c = 0$, we can use the expanded form of the given product to find the solutions.
• Graphing Quadratic Functions: The expanded form of the given product can be used to graph quadratic functions. For example, if we have a quadratic function in the form of $f(x) = ax^2 + bx + c$, we can use the expanded form of the given product to find the x-intercepts and y-intercepts of the graph.
• Optimization Problems: The expanded form of the given product can be used to solve optimization problems. For example, if we have a problem that involves maximizing or minimizing a quadratic function, we can use the expanded form of the given product to find the maximum or minimum value.

Tips and Tricks

Here are a few tips and tricks to help you expand products of two binomials:

• Use the Distributive Property: The distributive property is a fundamental concept in algebra that helps in expanding products of two binomials. Make sure to use the distributive property to multiply each term in the first binomial by each term in the second binomial.
• Combine Like Terms: Combining like terms is an essential step in simplifying the expression. Make sure to combine the like terms to simplify the expression.
• Check Your Work: Always check your work to ensure that the expression is simplified correctly. Use a calculator or a computer algebra system to check your work.

Conclusion

In conclusion, expanding a product of two binomials is a crucial skill that helps in solving various mathematical problems. The given product, $(2x - 5)(2x + 6)$, is a quadratic expression that can be expanded using the distributive property. We simplified the expression by combining like terms and arrived at the final form of the expression, which is $4x^2 + 2x - 30$. This is a quadratic expression that can be used to solve various mathematical problems.

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Introduction

In our previous article, we learned how to expand the given product $(2x - 5)(2x + 6)$ using the distributive property. We simplified the expression by combining like terms and arrived at the final form of the expression, which is $4x^2 + 2x - 30$. In this article, we will answer some frequently asked questions related to expanding products of two binomials.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

This property can be applied to expand a product of two binomials.

Q: How do I expand a product of two binomials?

A: To expand a product of two binomials, you need to multiply each term in the first binomial by each term in the second binomial. Then, combine like terms to simplify the expression.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $4x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $2x + 4x$, you can combine the like terms by adding the coefficients: $2x + 4x = 6x$.

Q: What is the final form of the expression $(2x - 5)(2x + 6)$?

A: The final form of the expression $(2x - 5)(2x + 6)$ is $4x^2 + 2x - 30$.

Q: How do I use the expanded form of the given product to solve quadratic equations?

A: The expanded form of the given product can be used to solve quadratic equations. For example, if you have a quadratic equation in the form of $ax^2 + bx + c = 0$, you can use the expanded form of the given product to find the solutions.

Q: How do I use the expanded form of the given product to graph quadratic functions?

A: The expanded form of the given product can be used to graph quadratic functions. For example, if you have a quadratic function in the form of $f(x) = ax^2 + bx + c$, you can use the expanded form of the given product to find the x-intercepts and y-intercepts of the graph.

Q: How do I use the expanded form of the given product to solve optimization problems?

A: The expanded form of the given product can be used to solve optimization problems. For example, if you have a problem that involves maximizing or minimizing a quadratic function, you can use the expanded form of the given product to find the maximum or minimum value.

Conclusion

In conclusion, expanding a product of two binomials is a crucial skill that helps in solving various mathematical problems. The given product, $(2x - 5)(2x + 6)$, is a quadratic expression that can be expanded using the distributive property. We simplified the expression by combining like terms and arrived at the final form of the expression, which is $4x^2 + 2x - 30$. This is a quadratic expression that can be used to solve various mathematical problems. We hope that this article has helped you to understand how to expand products of two binomials and how to use the expanded form to solve various mathematical problems.

Tips and Tricks

Here are a few tips and tricks to help you expand products of two binomials:

• Use the Distributive Property: The distributive property is a fundamental concept in algebra that helps in expanding products of two binomials. Make sure to use the distributive property to multiply each term in the first binomial by each term in the second binomial.
• Combine Like Terms: Combining like terms is an essential step in simplifying the expression. Make sure to combine the like terms to simplify the expression.
• Check Your Work: Always check your work to ensure that the expression is simplified correctly. Use a calculator or a computer algebra system to check your work.

Examples and Applications

The expanded form of the given product has various applications in mathematics and real-world problems. Here are a few examples:

• Quadratic Equations: The expanded form of the given product can be used to solve quadratic equations. For example, if we have a quadratic equation in the form of $ax^2 + bx + c = 0$, we can use the expanded form of the given product to find the solutions.
• Graphing Quadratic Functions: The expanded form of the given product can be used to graph quadratic functions. For example, if we have a quadratic function in the form of $f(x) = ax^2 + bx + c$, we can use the expanded form of the given product to find the x-intercepts and y-intercepts of the graph.
• Optimization Problems: The expanded form of the given product can be used to solve optimization problems. For example, if we have a problem that involves maximizing or minimizing a quadratic function, we can use the expanded form of the given product to find the maximum or minimum value.