Use The Graph To Find The Factorization Of $x^2 - 5x + 4$.A. $(x - 1)(x - 4)$ B. \$(x + 2)(x + 4)$[/tex\] C. $(x + 1)(x + 4)$ D. $(x + 2)(x - 2)$

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Introduction

Factorizing quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. In this article, we will explore the process of factorizing quadratic expressions using the graph method. We will use the given quadratic expression $x^2 - 5x + 4$ as an example to demonstrate the step-by-step process.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our example, the quadratic expression is $x^2 - 5x + 4$, where $a = 1$, $b = -5$, and $c = 4$.

The Graph Method

The graph method is a visual approach to factorizing quadratic expressions. It involves plotting the graph of the quadratic expression and identifying the x-intercepts, which represent the factors of the expression. To use the graph method, we need to follow these steps:

Step 1: Plot the Graph

To plot the graph of the quadratic expression $x^2 - 5x + 4$, we need to find the x-intercepts by setting the expression equal to zero and solving for x. This will give us the points where the graph intersects the x-axis.

Step 2: Identify the Factors

Once we have plotted the graph, we need to identify the factors of the quadratic expression. The factors are the values of x that make the expression equal to zero. In our example, the x-intercepts are x = 1 and x = 4, which represent the factors (x - 1) and (x - 4).

Applying the Graph Method

Now that we have understood the graph method, let's apply it to our example quadratic expression $x^2 - 5x + 4$. We will follow the steps outlined above to factorize the expression.

Step 1: Plot the Graph

To plot the graph of the quadratic expression $x^2 - 5x + 4$, we need to find the x-intercepts by setting the expression equal to zero and solving for x. This will give us the points where the graph intersects the x-axis.

import sympy as sp

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

# Define the quadratic expression
expr = x**2 - 5*x + 4

# Solve for x
solutions = sp.solve(expr, x)

print(solutions)

The output of the code above will give us the x-intercepts, which are x = 1 and x = 4.

Step 2: Identify the Factors

Once we have plotted the graph, we need to identify the factors of the quadratic expression. The factors are the values of x that make the expression equal to zero. In our example, the x-intercepts are x = 1 and x = 4, which represent the factors (x - 1) and (x - 4).

Conclusion

In this article, we have explored the process of factorizing quadratic expressions using the graph method. We have used the given quadratic expression $x^2 - 5x + 4$ as an example to demonstrate the step-by-step process. By following the steps outlined above, we have identified the factors of the quadratic expression as (x - 1) and (x - 4).

Answer

The correct answer is:

  • A. (x - 1)(x - 4)

This is the factorization of the quadratic expression $x^2 - 5x + 4$ using the graph method.

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Introduction

In our previous article, we explored the process of factorizing quadratic expressions using the graph method. We used the given quadratic expression $x^2 - 5x + 4$ as an example to demonstrate the step-by-step process. In this article, we will answer some frequently asked questions about factorizing quadratic expressions.

Q&A

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

A: Factorizing a quadratic expression involves expressing it as a product of two or more linear expressions, while simplifying a quadratic expression involves rewriting it in a simpler form, such as combining like terms.

Q: How do I determine if a quadratic expression can be factored?

A: To determine if a quadratic expression can be factored, you need to check if it can be expressed as a product of two or more linear expressions. You can use the graph method or the quadratic formula to help you determine if the expression can be factored.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that helps you find the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a$, $b$, and $c$ are the coefficients of the quadratic expression.

Q: How do I use the quadratic formula to factor a quadratic expression?

A: To use the quadratic formula to factor a quadratic expression, you need to follow these steps:

  1. Write down the quadratic expression in the form $ax^2 + bx + c$.
  2. Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
  3. Simplify the expression to find the solutions.

Q: What are the common mistakes to avoid when factorizing quadratic expressions?

A: Some common mistakes to avoid when factorizing quadratic expressions include:

  • Not checking if the expression can be factored before attempting to factor it.
  • Not using the correct method to factor the expression, such as using the graph method or the quadratic formula.
  • Not simplifying the expression after factoring it.

Q: How do I check if a quadratic expression is factorable?

A: To check if a quadratic expression is factorable, you need to follow these steps:

  1. Write down the quadratic expression in the form $ax^2 + bx + c$.
  2. Check if the expression can be expressed as a product of two or more linear expressions.
  3. Use the graph method or the quadratic formula to help you determine if the expression can be factored.

Q: What are the benefits of factorizing quadratic expressions?

A: Some benefits of factorizing quadratic expressions include:

  • Simplifying complex equations and solving problems more efficiently.
  • Identifying the x-intercepts and the factors of the expression.
  • Using the graph method or the quadratic formula to help you determine if the expression can be factored.

Conclusion

In this article, we have answered some frequently asked questions about factorizing quadratic expressions. We have covered topics such as the difference between factorizing and simplifying a quadratic expression, how to determine if a quadratic expression can be factored, and the benefits of factorizing quadratic expressions. By following the steps outlined above, you can become more confident in your ability to factorize quadratic expressions.

Additional Resources

For more information on factorizing quadratic expressions, you can check out the following resources:

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

We hope this article has been helpful in answering your questions about factorizing quadratic expressions. If you have any further questions or need additional help, please don't hesitate to ask.