Consider The Expression − 3 ( Y − 5 ) 2 − 9 + 7 Y -3(y-5)^2-9+7y − 3 ( Y − 5 ) 2 − 9 + 7 Y . Which Statements Are True About The Process And The Simplified Product? Check All That Apply.- The First Step In Simplifying Is To Distribute The -3 Throughout The Parentheses.- There Are 3 Terms In

Understanding the Expression

The given expression is $-3(y-5)^2-9+7y$. This expression involves a quadratic term, a constant term, and a linear term. To simplify this expression, we need to follow a specific order of operations.

Order of Operations

The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the Expression

To simplify the expression $-3(y-5)^2-9+7y$, we need to follow the order of operations.

Step 1: Evaluate the Expression Inside the Parentheses

The first step is to evaluate the expression inside the parentheses, which is $(y-5)^2$. This is a squared term, and we need to expand it.

(y-5)^2 = y^2 - 10y + 25

Step 2: Multiply the Coefficient by the Expanded Term

Now that we have expanded the squared term, we can multiply the coefficient $-3$ by the expanded term.

-3(y-5)^2 = -3(y^2 - 10y + 25)

Using the distributive property, we can expand the expression.

-3(y^2 - 10y + 25) = -3y^2 + 30y - 75

Step 3: Combine Like Terms

Now that we have expanded the squared term, we can combine like terms.

-3y^2 + 30y - 75 - 9 + 7y

Combining like terms, we get:

-3y^2 + 37y - 84

True Statements

Based on the steps above, the following statements are true about the process and the simplified product:

• The first step in simplifying is to distribute the -3 throughout the parentheses. This is true, as we first expanded the squared term and then multiplied the coefficient by the expanded term.
• There are 3 terms in the simplified product. This is true, as we have three terms in the simplified expression: $-3y^2$, $37y$, and $-84$.
• The simplified product is a quadratic expression. This is true, as the simplified expression is a quadratic expression in the form $ax^2 + bx + c$.
• The coefficient of the squared term is negative. This is true, as the coefficient of the squared term is $-3$.
• The constant term is negative. This is true, as the constant term is $-84$.

False Statements

Based on the steps above, the following statements are false about the process and the simplified product:

• The first step in simplifying is to combine like terms. This is false, as we first expanded the squared term and then multiplied the coefficient by the expanded term.
• There are 4 terms in the simplified product. This is false, as we have three terms in the simplified expression.
• The simplified product is a linear expression. This is false, as the simplified expression is a quadratic expression.
• The coefficient of the squared term is positive. This is false, as the coefficient of the squared term is $-3$.
• The constant term is positive. This is false, as the constant term is $-84$.

Conclusion

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Frequently Asked Questions

Q: What is the first step in simplifying a quadratic expression?

A: The first step in simplifying a quadratic expression is to evaluate the expression inside the parentheses. This may involve expanding a squared term or simplifying a complex expression.

Q: How do I expand a squared term?

A: To expand a squared term, you need to multiply each term inside the parentheses by itself. For example, if you have the expression $(x+2)^2$, you would expand it as follows:

(x+2)^2 = x^2 + 4x + 4

Q: What is the distributive property?

A: The distributive property is a rule that allows you to multiply a single term by multiple terms inside parentheses. For example, if you have the expression $-3(x+2)$, you would multiply the $-3$ by each term inside the parentheses:

-3(x+2) = -3x - 6

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract terms that have the same variable and exponent. For example, if you have the expression $2x + 3x$, you would combine the like terms as follows:

2x + 3x = 5x

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

A: A quadratic expression is an expression that contains a squared term, while a linear expression is an expression that contains only a single variable term. For example, the expression $x^2 + 3x$ is a quadratic expression, while the expression $2x + 3$ is a linear expression.

Q: How do I determine the coefficient of a squared term?

A: To determine the coefficient of a squared term, you need to look at the number that is multiplied by the squared term. For example, if you have the expression $-3x^2$, the coefficient of the squared term is $-3$.

Q: What is the constant term in a quadratic expression?

A: The constant term in a quadratic expression is the term that does not contain any variables. For example, in the expression $x^2 + 3x + 4$, the constant term is $4$.

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

A: To simplify a quadratic expression with a negative coefficient, you need to follow the same steps as you would for a positive coefficient. However, you will need to be careful when multiplying and dividing negative numbers.

Q: What is the final answer to the expression $-3(y-5)^2-9+7y$?

A: The final answer to the expression $-3(y-5)^2-9+7y$ is $-3y^2 + 37y - 84$.

Common Mistakes

Mistake 1: Not evaluating the expression inside the parentheses first

A: When simplifying a quadratic expression, it is essential to evaluate the expression inside the parentheses first. This will help you avoid mistakes and ensure that you are simplifying the expression correctly.

Mistake 2: Not using the distributive property

A: The distributive property is a crucial rule in algebra that allows you to multiply a single term by multiple terms inside parentheses. Failing to use the distributive property can lead to mistakes and incorrect simplifications.

Mistake 3: Not combining like terms

A: Combining like terms is an essential step in simplifying a quadratic expression. Failing to combine like terms can lead to mistakes and incorrect simplifications.

Conclusion

In conclusion, simplifying quadratic expressions requires careful attention to detail and a thorough understanding of the rules of algebra. By following the steps outlined in this guide, you can simplify quadratic expressions with ease and avoid common mistakes. Remember to evaluate the expression inside the parentheses first, use the distributive property, and combine like terms to ensure that you are simplifying the expression correctly.