Simplify The Expression:$9x^2 - 51x + 30$

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves rewriting an expression in a more compact and manageable form, often by combining like terms. In this article, we will simplify the given expression: 9x2−51x+309x^2 - 51x + 30. We will use various techniques, including factoring and combining like terms, to arrive at the simplified form.

Understanding the Expression

The given expression is a quadratic expression in the form of ax2+bx+cax^2 + bx + c, where a=9a = 9, b=−51b = -51, and c=30c = 30. To simplify this expression, we need to examine its structure and identify any patterns or relationships between the terms.

Factoring the Expression

One approach to simplifying the expression is to factor it. Factoring involves expressing the expression as a product of two or more simpler expressions. In this case, we can try to factor the expression by finding two numbers whose product is 9×30=2709 \times 30 = 270 and whose sum is −51-51.

# Factoring the Expression

## Step 1: Find two numbers whose product is 270 and whose sum is -51

We can start by listing the factors of 270 and checking if any pair of factors adds up to -51.

## Step 2: Factor the expression

Once we find the correct pair of factors, we can rewrite the expression as a product of two simpler expressions.

## Step 3: Simplify the expression

After factoring the expression, we can simplify it further by combining like terms.

Combining Like Terms

Another approach to simplifying the expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we can combine the terms 9x29x^2 and −51x-51x by adding their coefficients.

# Combining Like Terms

## Step 1: Identify the like terms

The like terms in the expression are $9x^2$ and $-51x$.

## Step 2: Combine the like terms

We can combine the like terms by adding their coefficients.

## Step 3: Simplify the expression

After combining the like terms, we can simplify the expression further by factoring out any common factors.

Simplifying the Expression

Now that we have factored and combined like terms, we can simplify the expression further by factoring out any common factors. In this case, we can factor out a common factor of 3 from the terms 9x29x^2 and −51x-51x.

# Simplifying the Expression

## Step 1: Factor out a common factor of 3

We can factor out a common factor of 3 from the terms $9x^2$ and $-51x$.

## Step 2: Simplify the expression

After factoring out the common factor, we can simplify the expression further by combining the remaining terms.

Conclusion

In this article, we simplified the expression 9x2−51x+309x^2 - 51x + 30 using various techniques, including factoring and combining like terms. We factored the expression by finding two numbers whose product is 9×30=2709 \times 30 = 270 and whose sum is −51-51. We then combined like terms by adding the coefficients of the terms 9x29x^2 and −51x-51x. Finally, we simplified the expression further by factoring out a common factor of 3 from the terms 9x29x^2 and −51x-51x. The simplified expression is 3(3x2−17x+10)3(3x^2 - 17x + 10).

Final Answer

Introduction

In our previous article, we simplified the expression 9x2−51x+309x^2 - 51x + 30 using various techniques, including factoring and combining like terms. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to examine its structure and identify any patterns or relationships between the terms. This involves looking for like terms, factoring, and other techniques that can help simplify the expression.

Q: How do I factor an expression?

A: Factoring an expression involves expressing it as a product of two or more simpler expressions. To factor an expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the difference between factoring and combining like terms?

A: Factoring involves expressing an expression as a product of two or more simpler expressions, while combining like terms involves adding or subtracting terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms that have the same variable raised to the same power.

Q: What is the final answer to the expression 9x2−51x+309x^2 - 51x + 30?

A: The final answer to the expression 9x2−51x+309x^2 - 51x + 30 is 3(3x2−17x+10)\boxed{3(3x^2 - 17x + 10)}.

Q: Can I simplify an expression by using a calculator?

A: While a calculator can be a useful tool for simplifying expressions, it is not always the best approach. Simplifying expressions by hand can help you understand the underlying math and can be a more efficient way to simplify complex expressions.

Q: How do I know if an expression can be simplified?

A: An expression can be simplified if it contains like terms or if it can be factored into simpler expressions. You can use various techniques, such as factoring and combining like terms, to simplify an expression.

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

A: Some common mistakes to avoid when simplifying expressions include:

  • Not combining like terms
  • Not factoring expressions that can be factored
  • Not simplifying expressions that can be simplified
  • Using a calculator to simplify expressions without understanding the underlying math

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions. We discussed the first step in simplifying an expression, factoring, combining like terms, and other techniques that can help simplify expressions. We also provided some common mistakes to avoid when simplifying expressions.

Final Answer

The final answer is 3(3x2−17x+10)\boxed{3(3x^2 - 17x + 10)}.