Simplify: { -3(y+2)^2 - 5 + 6y$} W H A T I S T H E S I M P L I F I E D P R O D U C T I N S T A N D A R D F O R M ? What Is The Simplified Product In Standard Form? Wha T I S T H Es Im Pl I F I E D P Ro D U C T In S T An D A R Df Or M ? { \square \, Y^2 + \square \, Y + \square\$}

Introduction

Quadratic expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will focus on simplifying the given expression: {-3(y+2)^2 - 5 + 6y$}$. We will break down the process into manageable steps, using a combination of algebraic manipulations and mathematical properties to arrive at the simplified product in standard form.

Step 1: Expand the Squared Term

The first step in simplifying the given expression is to expand the squared term: {-3(y+2)^2$}$. To do this, we will use the formula for expanding a squared binomial: ${(a+b)^2 = a^2 + 2ab + b^2\$}$. In this case, {a = y$}$ and {b = 2$}$.

-3(y+2)^2 = -3(y^2 + 4y + 4)

Step 2: Simplify the Expression

Now that we have expanded the squared term, we can simplify the expression by combining like terms. We will start by distributing the negative sign to each term inside the parentheses:

-3(y^2 + 4y + 4) = -3y^2 - 12y - 12

Next, we will add the remaining terms: ${-5 + 6y\$}$. To do this, we will combine the constant terms and the terms with the variable {y$}$:

-3y^2 - 12y - 12 - 5 + 6y = -3y^2 - 6y - 17

Step 3: Write the Simplified Product in Standard Form

The final step is to write the simplified product in standard form. To do this, we will rearrange the terms in descending order of the exponent of {y$}$:

-3y^2 - 6y - 17

Conclusion

In this article, we have simplified the given expression: {-3(y+2)^2 - 5 + 6y$}$. We have broken down the process into manageable steps, using a combination of algebraic manipulations and mathematical properties to arrive at the simplified product in standard form. The final answer is: ${-3y^2 - 6y - 17\$}$.

Key Takeaways

• To simplify a quadratic expression, we need to expand any squared terms and combine like terms.
• We can use the formula for expanding a squared binomial to simplify the expression.
• We can combine constant terms and terms with the variable {y$}$ to simplify the expression.
• We can write the simplified product in standard form by rearranging the terms in descending order of the exponent of {y$}$.

Frequently Asked Questions

• Q: What is the simplified product in standard form? A: The simplified product in standard form is: ${-3y^2 - 6y - 17\$}$.
• Q: How do I simplify a quadratic expression? A: To simplify a quadratic expression, you need to expand any squared terms and combine like terms.
• Q: What is the formula for expanding a squared binomial? A: The formula for expanding a squared binomial is: ${(a+b)^2 = a^2 + 2ab + b^2\$}$.

Additional Resources

• For more information on simplifying quadratic expressions, please refer to the following resources:

• Khan Academy: Simplifying Quadratic Expressions
• Mathway: Simplifying Quadratic Expressions
• Wolfram Alpha: Simplifying Quadratic Expressions
  Simplifying Quadratic Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the process of simplifying quadratic expressions. In this article, we will delve deeper into the topic and provide a comprehensive Q&A guide to help you master the art of simplifying quadratic expressions.

Q: What is a quadratic expression?

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

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

A: A quadratic expression is a polynomial expression of degree two, while a quadratic equation is a quadratic expression that is set equal to zero. For example, ${x^2 + 4x + 4 = 0\$}$ is a quadratic equation, while ${x^2 + 4x + 4\$}$ is a quadratic expression.

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, you need to expand any squared terms and combine like terms. You can use the formula for expanding a squared binomial: ${(a+b)^2 = a^2 + 2ab + b^2\$}$. Then, combine the constant terms and the terms with the variable.

Q: What is the formula for expanding a squared binomial?

A: The formula for expanding a squared binomial is: ${(a+b)^2 = a^2 + 2ab + b^2\$}$.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable. For example, if you have the expression: ${2x + 3x + 4\$}$, you can combine the like terms by adding the coefficients: ${5x + 4\$}$.

Q: What is the difference between a monomial and a polynomial?

A: A monomial is a single term, while a polynomial is a sum of monomials. For example, ${x^2\$}$ is a monomial, while ${x^2 + 3x + 4\$}$ is a polynomial.

Q: How do I write a quadratic expression in standard form?

A: To write a quadratic expression in standard form, you need to rearrange the terms in descending order of the exponent of the variable. For example, if you have the expression: ${3x^2 + 2x - 4\$}$, you can write it in standard form by rearranging the terms: ${3x^2 + 2x - 4\$}$.

Q: What is the importance of simplifying quadratic expressions?

A: Simplifying quadratic expressions is important because it helps to:

• Make the expression easier to work with
• Simplify the solution to a quadratic equation
• Make it easier to graph the quadratic function

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to:

• Find the x-intercepts of the function
• Find the vertex of the function
• Plot the points on a coordinate plane

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you master the art of simplifying quadratic expressions. We have covered topics such as the difference between a quadratic expression and a quadratic equation, how to simplify a quadratic expression, and how to write a quadratic expression in standard form. We hope that this guide has been helpful in your understanding of quadratic expressions.

Additional Resources

• For more information on simplifying quadratic expressions, please refer to the following resources:

• Khan Academy: Simplifying Quadratic Expressions
• Mathway: Simplifying Quadratic Expressions
• Wolfram Alpha: Simplifying Quadratic Expressions

Frequently Asked Questions

• Q: What is a quadratic expression? A: A quadratic expression is a polynomial expression of degree two.
• Q: How do I simplify a quadratic expression? A: To simplify a quadratic expression, you need to expand any squared terms and combine like terms.
• Q: What is the formula for expanding a squared binomial? A: The formula for expanding a squared binomial is: ${(a+b)^2 = a^2 + 2ab + b^2\$}$.

Key Takeaways

• A quadratic expression is a polynomial expression of degree two.
• To simplify a quadratic expression, you need to expand any squared terms and combine like terms.
• The formula for expanding a squared binomial is: ${(a+b)^2 = a^2 + 2ab + b^2\$}$.
• To write a quadratic expression in standard form, you need to rearrange the terms in descending order of the exponent of the variable.