Simplify The Expression: { (2x - 9)(3x - 5)$}$

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps in solving equations and inequalities. It involves combining like terms, removing parentheses, and rearranging the expression to make it easier to work with. In this article, we will simplify the expression $(2x - 9)(3x - 5)$ using the distributive property and combining like terms.

The Distributive Property

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The distributive property is a fundamental concept in algebra that allows us to expand expressions with parentheses. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

We can use this property to simplify the expression $(2x - 9)(3x - 5)$ by multiplying each term inside the parentheses with the other term.

Expanding the Expression

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To expand the expression, we multiply each term inside the first parentheses with each term inside the second parentheses:

$$(2x - 9)(3x - 5) = (2x)(3x) + (2x)(-5) + (-9)(3x) + (-9)(-5)$$

Simplifying the Expression

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Now, we simplify each term by combining like terms and removing parentheses:

$$(2x)(3x) = 6x^2$$

$$(2x)(-5) = -10x$$

$$(-9)(3x) = -27x$$

$$(-9)(-5) = 45$$

Combining Like Terms

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We can now combine like terms by adding or subtracting the coefficients of the same variables:

$$6x^2 - 10x - 27x + 45$$

Final Simplification

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Combining the like terms, we get:

$$6x^2 - 37x + 45$$

Conclusion

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In this article, we simplified the expression $(2x - 9)(3x - 5)$ using the distributive property and combining like terms. We expanded the expression by multiplying each term inside the parentheses with the other term, and then simplified each term by combining like terms and removing parentheses. The final simplified expression is $6x^2 - 37x + 45$.

Example Use Cases

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Simplifying expressions like $(2x - 9)(3x - 5)$ is essential in solving equations and inequalities. Here are a few example use cases:

• Solving quadratic equations: Simplifying expressions like $(2x - 9)(3x - 5)$ can help us solve quadratic equations of the form $ax^2 + bx + c = 0$.
• Factoring expressions: Simplifying expressions like $(2x - 9)(3x - 5)$ can help us factor expressions of the form $ax^2 + bx + c$.
• Solving systems of equations: Simplifying expressions like $(2x - 9)(3x - 5)$ can help us solve systems of equations of the form $ax^2 + bx + c = 0$ and $dx^2 + ex + f = 0$.

Tips and Tricks

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Here are a few tips and tricks to help you simplify expressions like $(2x - 9)(3x - 5)$:

• Use the distributive property to expand expressions with parentheses.
• Combine like terms by adding or subtracting the coefficients of the same variables.
• Remove parentheses by multiplying each term inside the parentheses with the other term.
• Simplify each term by combining like terms and removing parentheses.

Common Mistakes

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Here are a few common mistakes to avoid when simplifying expressions like $(2x - 9)(3x - 5)$:

• Failing to use the distributive property to expand expressions with parentheses.
• Failing to combine like terms by adding or subtracting the coefficients of the same variables.
• Failing to remove parentheses by multiplying each term inside the parentheses with the other term.
• Failing to simplify each term by combining like terms and removing parentheses.

Conclusion

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In conclusion, simplifying expressions like $(2x - 9)(3x - 5)$ is an essential skill in algebra that helps us solve equations and inequalities. By using the distributive property, combining like terms, and removing parentheses, we can simplify expressions and make them easier to work with. Remember to use the distributive property to expand expressions with parentheses, combine like terms by adding or subtracting the coefficients of the same variables, remove parentheses by multiplying each term inside the parentheses with the other term, and simplify each term by combining like terms and removing parentheses.

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Introduction

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In our previous article, we simplified the expression $(2x - 9)(3x - 5)$ using the distributive property and combining like terms. In this article, we will answer some frequently asked questions about simplifying expressions like $(2x - 9)(3x - 5)$.

Q&A

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Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions with parentheses. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

Q: How do I use the distributive property to simplify expressions like $(2x - 9)(3x - 5)$?

A: To use the distributive property, multiply each term inside the first parentheses with each term inside the second parentheses:

$$(2x - 9)(3x - 5) = (2x)(3x) + (2x)(-5) + (-9)(3x) + (-9)(-5)$$

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

A: Combining like terms involves adding or subtracting the coefficients of the same variables, while removing parentheses involves multiplying each term inside the parentheses with the other term.

Q: How do I combine like terms in an expression like $(2x - 9)(3x - 5)$?

A: To combine like terms, add or subtract the coefficients of the same variables:

$$6x^2 - 10x - 27x + 45$$

Q: What is the final simplified expression for $(2x - 9)(3x - 5)$?

A: The final simplified expression is $6x^2 - 37x + 45$.

Q: What are some common mistakes to avoid when simplifying expressions like $(2x - 9)(3x - 5)$?

A: Some common mistakes to avoid include:

• Failing to use the distributive property to expand expressions with parentheses.
• Failing to combine like terms by adding or subtracting the coefficients of the same variables.
• Failing to remove parentheses by multiplying each term inside the parentheses with the other term.
• Failing to simplify each term by combining like terms and removing parentheses.

Q: How do I use the distributive property to simplify expressions with multiple parentheses?

A: To use the distributive property with multiple parentheses, multiply each term inside the first parentheses with each term inside the second parentheses, and then multiply each term inside the resulting expression with each term inside the third parentheses, and so on.

Q: What are some example use cases for simplifying expressions like $(2x - 9)(3x - 5)$?

A: Some example use cases include:

• Solving quadratic equations: Simplifying expressions like $(2x - 9)(3x - 5)$ can help us solve quadratic equations of the form $ax^2 + bx + c = 0$.
• Factoring expressions: Simplifying expressions like $(2x - 9)(3x - 5)$ can help us factor expressions of the form $ax^2 + bx + c$.
• Solving systems of equations: Simplifying expressions like $(2x - 9)(3x - 5)$ can help us solve systems of equations of the form $ax^2 + bx + c = 0$ and $dx^2 + ex + f = 0$.

Conclusion

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In conclusion, simplifying expressions like $(2x - 9)(3x - 5)$ is an essential skill in algebra that helps us solve equations and inequalities. By using the distributive property, combining like terms, and removing parentheses, we can simplify expressions and make them easier to work with. Remember to use the distributive property to expand expressions with parentheses, combine like terms by adding or subtracting the coefficients of the same variables, remove parentheses by multiplying each term inside the parentheses with the other term, and simplify each term by combining like terms and removing parentheses.

Tips and Tricks

---

Here are a few tips and tricks to help you simplify expressions like $(2x - 9)(3x - 5)$:

• Use the distributive property to expand expressions with parentheses.
• Combine like terms by adding or subtracting the coefficients of the same variables.
• Remove parentheses by multiplying each term inside the parentheses with the other term.
• Simplify each term by combining like terms and removing parentheses.

Common Mistakes

---

Here are a few common mistakes to avoid when simplifying expressions like $(2x - 9)(3x - 5)$:

• Failing to use the distributive property to expand expressions with parentheses.
• Failing to combine like terms by adding or subtracting the coefficients of the same variables.
• Failing to remove parentheses by multiplying each term inside the parentheses with the other term.
• Failing to simplify each term by combining like terms and removing parentheses.