Simplify The Following Expression:$x^2 - 9x + 20 \div (x - 4$\]

Introduction

In this article, we will simplify the given expression: $x^2 - 9x + 20 \div (x - 4)$. This involves breaking down the expression into manageable parts, applying the order of operations, and simplifying the resulting expression. We will use algebraic techniques and mathematical concepts to simplify the expression.

Understanding the Expression

The given expression is a quadratic expression divided by a linear expression. The quadratic expression is $x^2 - 9x + 20$, and the linear expression is $x - 4$. To simplify the expression, we need to apply the order of operations, which states that we should perform division before subtraction.

Step 1: Factor the Quadratic Expression

The quadratic expression $x^2 - 9x + 20$ can be factored as $(x - 4)(x - 5)$. This is a difference of squares, where $a^2 - b^2 = (a - b)(a + b)$. In this case, $a = x$ and $b = 5$.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the quadratic expression
quadratic_expression = x**2 - 9*x + 20

# Factor the quadratic expression
factored_expression = sp.factor(quadratic_expression)

print(factored_expression)

Step 2: Simplify the Expression

Now that we have factored the quadratic expression, we can simplify the expression by canceling out the common factor. The expression becomes:

$$\frac{(x - 4)(x - 5)}{x - 4}$$

We can cancel out the common factor $(x - 4)$, which leaves us with:

$$x - 5$$

Step 3: Check for Extraneous Solutions

When we simplified the expression, we canceled out the common factor $(x - 4)$. However, this means that the original expression is undefined when $x = 4$, because division by zero is undefined. Therefore, we need to check if $x = 4$ is a solution to the original expression.

Substituting $x = 4$ into the original expression, we get:

$$4^2 - 9(4) + 20 \div (4 - 4)$$

This expression is undefined, because division by zero is undefined. Therefore, $x = 4$ is not a solution to the original expression.

Conclusion

In this article, we simplified the given expression $x^2 - 9x + 20 \div (x - 4)$ by factoring the quadratic expression and canceling out the common factor. We also checked for extraneous solutions and found that $x = 4$ is not a solution to the original expression. The simplified expression is $x - 5$.

Final Answer

The final answer is $\boxed{x - 5}$.

Additional Resources

For more information on simplifying expressions, see the following resources:

• Algebraic Techniques for Simplifying Expressions
• Order of Operations
• Factoring Quadratic Expressions

Related Articles

• Simplifying Rational Expressions
• Simplifying Radical Expressions
• Simplifying Exponential Expressions
  Simplify the Given Expression: A Q&A Guide ===========================================================

Introduction

In our previous article, we simplified the given expression: $x^2 - 9x + 20 \div (x - 4)$. We factored the quadratic expression, canceled out the common factor, and checked for extraneous solutions. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the order of operations?

A: 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.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the resulting expression.

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

A: A quadratic expression is a polynomial expression of degree two, which means it has a squared variable. A linear expression is a polynomial expression of degree one, which means it has a variable but no squared variable.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to substitute the value of the variable into the original expression and see if it is defined. If the expression is undefined, then the value is an extraneous solution.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $x - 5$.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to follow the same steps as before. However, you may need to use substitution or elimination methods to simplify the expression.

Q: What is the importance of simplifying expressions?

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

• Reduce the complexity of an expression
• Make it easier to solve equations and inequalities
• Understand the behavior of a function
• Make it easier to graph a function

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions. We covered topics such as the order of operations, simplifying rational expressions, and checking for extraneous solutions. We also discussed the importance of simplifying expressions and how it can help us to understand the behavior of a function.

Final Answer

The final answer is $\boxed{x - 5}$.

Additional Resources

For more information on simplifying expressions, see the following resources:

• Algebraic Techniques for Simplifying Expressions
• Order of Operations
• Factoring Quadratic Expressions

Related Articles

• Simplifying Rational Expressions
• Simplifying Radical Expressions
• Simplifying Exponential Expressions