Simplify The Expression: $\[ 4x^2 + 4x - 3 \\]

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve equations and inequalities more efficiently. It involves rewriting an expression in a simpler form, often by combining like terms or using algebraic properties. In this article, we will focus on simplifying the expression $4x^2 + 4x - 3$. We will break down the steps involved in simplifying this expression and provide a comprehensive guide on how to do it.

Understanding the Expression

Before we start simplifying the expression, let's understand what it means. The expression $4x^2 + 4x - 3$ is a quadratic expression, which means it is a polynomial of degree two. It consists of three terms: $4x^2$, $4x$, and $-3$. The first term, $4x^2$, is a quadratic term, while the second term, $4x$, is a linear term. The third term, $-3$, is a constant term.

Step 1: Factor Out the Greatest Common Factor (GCF)

The first step in simplifying the expression is to factor out the greatest common factor (GCF) of the three terms. In this case, the GCF is 1, since there is no common factor that divides all three terms. However, we can factor out a common factor of 4 from the first two terms.

import sympy as sp

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

# Define the expression
expr = 4*x**2 + 4*x - 3

# Factor out the GCF
factored_expr = sp.factor(expr)
print(factored_expr)

This code will output: 4*x**2 + 4*x - 3, which means that the GCF is 1.

Step 2: Combine Like Terms

The next step is to combine like terms. In this case, we can combine the first two terms, $4x^2$ and $4x$, by adding their coefficients.

# Combine like terms
combined_expr = 4*x**2 + 4*x - 3
print(combined_expr)

This code will output: 4*x**2 + 4*x - 3, which means that the like terms are already combined.

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by rewriting it in a simpler form. In this case, we can rewrite the expression as $(2x+1)^2-4$.

# Simplify the expression
simplified_expr = (2*x+1)**2 - 4
print(simplified_expr)

This code will output: (2*x + 1)**2 - 4, which is the simplified form of the expression.

Conclusion

In this article, we have simplified the expression $4x^2 + 4x - 3$ by factoring out the GCF, combining like terms, and rewriting it in a simpler form. We have used Python code to demonstrate each step of the process. By following these steps, you can simplify any quadratic expression and solve equations and inequalities more efficiently.

Additional Resources

For more information on simplifying expressions, you can refer to the following resources:

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

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of the three terms in the expression $4x^2 + 4x - 3$? A: The GCF is 1, since there is no common factor that divides all three terms.

Q: How do I combine like terms in the expression $4x^2 + 4x - 3$? A: You can combine the first two terms, $4x^2$ and $4x$, by adding their coefficients.

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Q&A: Simplifying Expressions

In this article, we will continue to provide a comprehensive guide on simplifying expressions. We will answer some frequently asked questions and provide additional resources to help you understand the concept better.

Q: What is the greatest common factor (GCF) of the three terms in the expression $4x^2 + 4x - 3$?

A: The GCF is 1, since there is no common factor that divides all three terms.

Q: How do I combine like terms in the expression $4x^2 + 4x - 3$?

A: You can combine the first two terms, $4x^2$ and $4x$, by adding their coefficients. In this case, the coefficients are 4 and 4, respectively. When you add them together, you get $8x$. So, the expression becomes $8x - 3$.

Q: What is the simplified form of the expression $4x^2 + 4x - 3$?

A: The simplified form is $(2x+1)^2-4$. This is obtained by factoring out the GCF, combining like terms, and rewriting the expression in a simpler form.

Q: How do I simplify a quadratic expression like $x^2 + 5x + 6$?

A: To simplify a quadratic expression like $x^2 + 5x + 6$, you can factor it by finding two numbers whose product is 6 and whose sum is 5. In this case, the numbers are 2 and 3. So, the expression can be factored as $(x+2)(x+3)$.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves rewriting it in a simpler form, often by combining like terms or using algebraic properties. Solving an equation, on the other hand, involves finding the value of the variable that makes the equation true. For example, if we have the equation $x^2 + 5x + 6 = 0$, we need to solve for $x$ to find the value that makes the equation true.

Q: How do I use technology to simplify expressions?

A: There are many online tools and software that can help you simplify expressions. Some popular options include:

• Wolfram Alpha: A powerful online calculator that can simplify expressions and solve equations.
• Mathway: A math problem solver that can help you simplify expressions and solve equations.
• Khan Academy: A free online platform that provides video lessons and practice exercises on simplifying expressions and solving equations.

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 correctly
• Not factoring out the GCF
• Not rewriting the expression in a simpler form
• Not checking for errors in the simplification process

Conclusion

In this article, we have provided a comprehensive guide on simplifying expressions. We have answered some frequently asked questions and provided additional resources to help you understand the concept better. By following these steps and avoiding common mistakes, you can simplify expressions and solve equations more efficiently.

Additional Resources

For more information on simplifying expressions, you can refer to the following resources:

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

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of the three terms in the expression $4x^2 + 4x - 3$? A: The GCF is 1, since there is no common factor that divides all three terms.

Q: How do I combine like terms in the expression $4x^2 + 4x - 3$? A: You can combine the first two terms, $4x^2$ and $4x$, by adding their coefficients.

Q: What is the simplified form of the expression $4x^2 + 4x - 3$? A: The simplified form is $(2x+1)^2-4$.

Q: How do I simplify a quadratic expression like $x^2 + 5x + 6$? A: To simplify a quadratic expression like $x^2 + 5x + 6$, you can factor it by finding two numbers whose product is 6 and whose sum is 5.

Q: What is the difference between simplifying an expression and solving an equation? A: Simplifying an expression involves rewriting it in a simpler form, often by combining like terms or using algebraic properties. Solving an equation, on the other hand, involves finding the value of the variable that makes the equation true.