Simplify The Expression:${ B^2 - 19b + 90 }$

Mar 13, 2025 by ADMIN 46 views

**Simplifying Algebraic Expressions: A Step-by-Step Guide**

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given expression: $b^2 - 19b + 90$ . We will break down the process into manageable steps, making it easy to understand and follow along.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we're dealing with. The given expression is a quadratic expression, which means it has a squared variable ( $b^2$ ) and a linear term ( $-19b$ ). The constant term is $90$ . Our goal is to simplify this expression by factoring it into its simplest form.

Factoring Quadratic Expressions

Factoring quadratic expressions involves finding two binomials whose product equals the original expression. In other words, we need to find two binomials that, when multiplied together, give us the original expression. To do this, we can use the factoring method, which involves finding two numbers whose product equals the constant term and whose sum equals the coefficient of the linear term.

Step 1: Find Two Numbers Whose Product Equals the Constant Term

The constant term in our expression is $90$ . We need to find two numbers whose product equals $90$ . These numbers are $10$ and $9$ , since $10 \times 9 = 90$ .

Step 2: Find Two Numbers Whose Sum Equals the Coefficient of the Linear Term

The coefficient of the linear term in our expression is $-19$ . We need to find two numbers whose sum equals $-19$ . These numbers are $-10$ and $-9$ , since $-10 + (-9) = -19$ .

Step 3: Write the Expression as a Product of Two Binomials

Now that we have found the two numbers whose product equals the constant term and whose sum equals the coefficient of the linear term, we can write the expression as a product of two binomials. The two binomials are $(b - 10)$ and $(b - 9)$ .

Step 4: Simplify the Expression

Now that we have written the expression as a product of two binomials, we can simplify it by multiplying the two binomials together. This gives us:

$(b - 10)(b - 9) = b^2 - 9b - 10b + 90$

Step 5: Combine Like Terms

Now that we have multiplied the two binomials together, we can combine like terms to simplify the expression further. This gives us:

$b^2 - 19b + 90$

Conclusion

In this article, we have simplified the given expression $b^2 - 19b + 90$ by factoring it into its simplest form. We have broken down the process into manageable steps, making it easy to understand and follow along. By following these steps, you can simplify any quadratic expression and make it easier to work with.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. For example, in physics, you may need to simplify expressions to describe the motion of objects. In engineering, you may need to simplify expressions to design and build complex systems. In economics, you may need to simplify expressions to model and analyze economic systems.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

Use the factoring method: The factoring method is a powerful tool for simplifying quadratic expressions. By finding two binomials whose product equals the original expression, you can simplify the expression and make it easier to work with.
Use the distributive property: The distributive property is a fundamental concept in algebra that allows you to multiply expressions by distributing the terms. By using the distributive property, you can simplify expressions and make them easier to work with.
Combine like terms: Combining like terms is an essential skill for simplifying algebraic expressions. By combining like terms, you can simplify expressions and make them easier to work with.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

Not using the factoring method: The factoring method is a powerful tool for simplifying quadratic expressions. By not using the factoring method, you may end up with a complicated expression that is difficult to work with.
Not combining like terms: Combining like terms is an essential skill for simplifying algebraic expressions. By not combining like terms, you may end up with a complicated expression that is difficult to work with.
Not using the distributive property: The distributive property is a fundamental concept in algebra that allows you to multiply expressions by distributing the terms. By not using the distributive property, you may end up with a complicated expression that is difficult to work with.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify any quadratic expression and make it easier to work with. Remember to use the factoring method, combine like terms, and use the distributive property to simplify expressions. With practice and patience, you can become proficient in simplifying algebraic expressions and make them easier to work with.

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Introduction

In our previous article, we discussed the process of simplifying algebraic expressions, focusing on the given expression $b^2 - 19b + 90$ . We broke down the process into manageable steps, making it easy to understand and follow along. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

Q: What is the purpose of simplifying algebraic expressions?

A: The purpose of simplifying algebraic expressions is to make them easier to work with. By simplifying expressions, you can make calculations and problem-solving more efficient and accurate.

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression when it is necessary to make calculations or problem-solving more efficient and accurate. For example, if you are working with a quadratic expression and need to find the roots, simplifying the expression can make it easier to solve.

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

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

Not using the factoring method
Not combining like terms
Not using the distributive property
Not checking for errors in calculations

Q: How do I use the factoring method to simplify algebraic expressions?

A: To use the factoring method, follow these steps:

Find two numbers whose product equals the constant term
Find two numbers whose sum equals the coefficient of the linear term
Write the expression as a product of two binomials
Simplify the expression by combining like terms

Q: What is the distributive property, and how do I use it to simplify algebraic expressions?

A: The distributive property is a fundamental concept in algebra that allows you to multiply expressions by distributing the terms. To use the distributive property, follow these steps:

Multiply each term in the first expression by each term in the second expression
Combine like terms to simplify the expression

Q: How do I combine like terms to simplify algebraic expressions?

A: To combine like terms, follow these steps:

Identify the like terms in the expression
Add or subtract the coefficients of the like terms
Simplify the expression by combining the like terms

Q: What are some real-world applications of simplifying algebraic expressions?

A: Some real-world applications of simplifying algebraic expressions include:

Physics: Simplifying expressions to describe the motion of objects
Engineering: Simplifying expressions to design and build complex systems
Economics: Simplifying expressions to model and analyze economic systems

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article and avoiding common mistakes, you can simplify any quadratic expression and make it easier to work with. Remember to use the factoring method, combine like terms, and use the distributive property to simplify expressions. With practice and patience, you can become proficient in simplifying algebraic expressions and make them easier to work with.

Additional Resources

For more information on simplifying algebraic expressions, check out the following resources:

Khan Academy: Algebraic Expressions
Mathway: Simplifying Algebraic Expressions
Wolfram Alpha: Algebraic Expressions

Practice Problems

Try simplifying the following algebraic expressions:

$x^2 + 5x + 6$
$y^2 - 3y - 4$
$z^2 + 2z + 1$

Use the factoring method, combine like terms, and use the distributive property to simplify the expressions.