Select The Correct Answer.Which Quadratic Expression Represents The Product Of These Factors { (2x+5)(7-4x)$}$?A. { -8x^2+34x-35$}$ B. { -8x^2-6x+35$}$ C. { -8x^2-34x+35$}$ D. { -8x^2+6x-35$}$

Introduction

Quadratic expressions are a fundamental concept in algebra, and understanding how to solve them is crucial for success in mathematics. In this article, we will explore how to select the correct answer when given a quadratic expression that represents the product of two factors. We will use the example of the product of ${(2x+5)(7-4x)}$ to demonstrate the step-by-step process.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form of ${ax^2+bx+c}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable. Quadratic expressions can be factored, expanded, or solved using various methods.

Factoring Quadratic Expressions

Factoring a quadratic expression involves expressing it as a product of two binomials. This can be done using various methods, including the FOIL method, which stands for First, Outer, Inner, Last. The FOIL method involves multiplying the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms.

Expanding Quadratic Expressions

Expanding a quadratic expression involves expressing it as a sum of terms. This can be done using the distributive property, which states that ${a(b+c)=ab+ac}$. When expanding a quadratic expression, we multiply each term in the first binomial by each term in the second binomial.

Solving the Example

Now that we have a basic understanding of quadratic expressions, let's apply this knowledge to the example given in the problem. We are asked to find the product of the factors ${(2x+5)(7-4x)}$. To do this, we will use the FOIL method to expand the expression.

Step 1: Multiply the First Terms

The first term in the first binomial is ${2x}$, and the first term in the second binomial is ${7}$. Multiplying these two terms gives us ${14x}$.

Step 2: Multiply the Outer Terms

The outer term in the first binomial is ${2x}$, and the outer term in the second binomial is ${-4x}$. Multiplying these two terms gives us ${-8x^2}$.

Step 3: Multiply the Inner Terms

The inner term in the first binomial is ${5}$, and the inner term in the second binomial is ${-4x}$. Multiplying these two terms gives us ${-20x}$.

Step 4: Multiply the Last Terms

The last term in the first binomial is ${5}$, and the last term in the second binomial is ${7}$. Multiplying these two terms gives us ${35}$.

Step 5: Combine the Terms

Now that we have multiplied all the terms, we can combine them to get the final expression. The expression is ${-8x^2-20x+35}$.

Selecting the Correct Answer

Now that we have the final expression, we can compare it to the options given in the problem. The correct answer is the one that matches our final expression.

Option A: ${-8x^2+34x-35}$

This option does not match our final expression, so it is not the correct answer.

Option B: ${-8x^2-6x+35}$

This option does not match our final expression, so it is not the correct answer.

Option C: ${-8x^2-34x+35}$

This option does not match our final expression, so it is not the correct answer.

Option D: ${-8x^2+6x-35}$

This option does not match our final expression, so it is not the correct answer.

However, we notice that the correct answer is not among the options given. This is because the correct answer is ${-8x^2-20x+35}$, which is not among the options.

Conclusion

In conclusion, solving quadratic expressions involves factoring, expanding, or solving using various methods. In this article, we used the example of the product of ${(2x+5)(7-4x)}$ to demonstrate the step-by-step process of expanding a quadratic expression. We also compared our final expression to the options given in the problem to select the correct answer. However, we found that the correct answer was not among the options given.

Final Answer

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Frequently Asked Questions

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form of ${ax^2+bx+c}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable.

Q: How do I factor a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomials. This can be done using various methods, including the FOIL method, which stands for First, Outer, Inner, Last.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomials. It involves multiplying the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms.

Q: How do I expand a quadratic expression?

A: Expanding a quadratic expression involves expressing it as a sum of terms. This can be done using the distributive property, which states that ${a(b+c)=ab+ac}$. When expanding a quadratic expression, we multiply each term in the first binomial by each term in the second binomial.

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

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while expanding a quadratic expression involves expressing it as a sum of terms.

Q: How do I solve a quadratic equation?

A: Solving a quadratic equation involves finding the values of the variable that make the equation true. This can be done using various methods, including factoring, the quadratic formula, and graphing.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by ${x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$, where ${a}$, ${b}$, and ${c}$ are the coefficients of the quadratic equation.

Q: How do I graph a quadratic equation?

A: Graphing a quadratic equation involves plotting the points on a coordinate plane and drawing a smooth curve through them. This can be done using various methods, including the use of a graphing calculator or software.

Q: What is the significance of quadratic expressions in real-life applications?

A: Quadratic expressions have numerous real-life applications, including physics, engineering, economics, and computer science. They are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of financial markets.

Q: How do I choose the correct method for solving a quadratic expression?

A: Choosing the correct method for solving a quadratic expression depends on the specific problem and the tools available. Some common methods include factoring, the quadratic formula, and graphing.

Q: What are some common mistakes to avoid when solving quadratic expressions?

A: Some common mistakes to avoid when solving quadratic expressions include:

• Not following the order of operations
• Not using the correct method for solving the equation
• Not checking the solutions for validity
• Not considering the context of the problem

Conclusion

In conclusion, quadratic expressions are a fundamental concept in algebra, and understanding how to solve them is crucial for success in mathematics. This Q&A article provides a comprehensive overview of the concepts and methods involved in solving quadratic expressions, including factoring, expanding, and solving using various methods. By following the tips and avoiding common mistakes, you can become proficient in solving quadratic expressions and apply them to real-life applications.