Select The Correct Answer.Which Expression Is Equivalent To The Given Expression?${ -5y^2 + 50y - 105 }$A. { (-5y + 21)(y - 5)$}$B. { -5(y + 21)(y + 1)$}$C. { (5y - 35)(-y + 7)$}$D. { -5(y - 3)(y - 7)$}$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of algebraic expression, namely the quadratic expression. We will explore the different methods of factoring quadratic expressions and provide a step-by-step guide on how to solve them.

What is a Quadratic Expression?

A quadratic expression is a polynomial expression of degree two, which means it has a highest power of two. It is typically 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 into the product of two binomials, which is a fundamental concept in algebra.

Factoring Quadratic Expressions

Factoring quadratic expressions involves expressing them as the product of two binomials. This can be done using various methods, including the factoring method, the quadratic formula, and the completing the square method. In this article, we will focus on the factoring method.

The Factoring Method

The factoring method involves expressing a quadratic expression as the product of two binomials. This can be done by finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Step 1: Identify the Constant Term and the Coefficient of the Linear Term

The constant term is the term that is not multiplied by the variable, while the coefficient of the linear term is the number that multiplies the variable. In the given expression $-5y^2 + 50y - 105$, the constant term is $-105$ and the coefficient of the linear term is $50$.

Step 2: Find Two Numbers Whose Product is Equal to the Constant Term

The product of the two numbers must be equal to the constant term, which is $-105$. We can find two numbers whose product is equal to $-105$ by listing the factors of $-105$. The factors of $-105$ are $-1$, $-3$, $-5$, $-7$, $-15$, $-21$, $-35$, and $-105$.

Step 3: Find Two Numbers Whose Sum is Equal to the Coefficient of the Linear Term

The sum of the two numbers must be equal to the coefficient of the linear term, which is $50$. We can find two numbers whose sum is equal to $50$ by listing the pairs of numbers that add up to $50$. The pairs of numbers that add up to $50$ are $-1$ and $51$, $-3$ and $53$, $-5$ and $55$, $-7$ and $57$, $-15$ and $65$, $-21$ and $71$, $-35$ and $85$, and $-105$ and $155$.

Step 4: Factor the Quadratic Expression

Once we have found the two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term, we can factor the quadratic expression. In this case, we can factor the expression $-5y^2 + 50y - 105$ as $(-5y + 21)(y - 5)$.

Conclusion

In conclusion, solving algebraic expressions is a crucial skill for students to master. In this article, we have explored the different methods of factoring quadratic expressions and provided a step-by-step guide on how to solve them. By following these steps, students can easily factor quadratic expressions and solve them.

Answer

The correct answer is:

• A. [$(-5y + 21)(y - 5)$]

This is because the expression $-5y^2 + 50y - 105$ can be factored as $(-5y + 21)(y - 5)$.

Discussion

This problem is a great example of how to factor quadratic expressions using the factoring method. The factoring method involves expressing a quadratic expression as the product of two binomials. This can be done by finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Tips and Tricks

Here are some tips and tricks to help you solve this problem:

• Make sure to identify the constant term and the coefficient of the linear term.
• Find two numbers whose product is equal to the constant term.
• Find two numbers whose sum is equal to the coefficient of the linear term.
• Factor the quadratic expression using the two numbers you found.

By following these tips and tricks, you can easily solve this problem and factor quadratic expressions using the factoring method.

Related Problems

Here are some related problems that you can try to solve:

• Factor the quadratic expression $2x^2 + 12x + 18$.
• Factor the quadratic expression $3y^2 - 12y + 15$.
• Factor the quadratic expression $4z^2 + 20z + 25$.

By solving these problems, you can practice your skills in factoring quadratic expressions and become more confident in your abilities.

Conclusion

In conclusion, solving algebraic expressions is a crucial skill for students to master. In this article, we have explored the different methods of factoring quadratic expressions and provided a step-by-step guide on how to solve them. By following these steps, students can easily factor quadratic expressions and solve them.