Simplify The Expression: ${ 9x^2 + 6x - 8 }$

Understanding the Problem

In this problem, we are given a quadratic expression in the form of $ax^2 + bx + c$. Our goal is to simplify this expression, which means we need to rewrite it in a more compact and manageable form. The given expression is $9x^2 + 6x - 8$, and we need to simplify it.

What is Simplification in Algebra?

Simplification in algebra refers to the process of rewriting an expression in a more compact and manageable form, without changing its value. This involves combining like terms, removing any unnecessary factors, and rearranging the terms to make the expression easier to work with.

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) from the terms. In this case, the GCF of $9x^2$, $6x$, and $-8$ is $1$, since there is no common factor that divides all three terms.

However, we can factor out a common factor of $3$ from the first two terms, which are $9x^2$ and $6x$. This gives us:

$$9x^2 + 6x - 8 = 3(3x^2 + 2x) - 8$$

Step 2: Simplify the Expression Inside the Parentheses

Now that we have factored out the common factor of $3$ from the first two terms, we can simplify the expression inside the parentheses. The expression inside the parentheses is $3x^2 + 2x$, which can be rewritten as:

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

So, the expression becomes:

$$9x^2 + 6x - 8 = 3(x(3x + 2)) - 8$$

Step 3: Distribute the Common Factor

The next step is to distribute the common factor of $3$ to the terms inside the parentheses. This gives us:

$$9x^2 + 6x - 8 = 3x(3x + 2) - 8$$

Step 4: Simplify the Expression

Now that we have distributed the common factor, we can simplify the expression further. The expression becomes:

$$9x^2 + 6x - 8 = 9x^2 + 6x - 8$$

The Final Answer

After simplifying the expression, we find that the final answer is:

$$9x^2 + 6x - 8$$

Conclusion

In this problem, we simplified the quadratic expression $9x^2 + 6x - 8$ by factoring out the greatest common factor (GCF), simplifying the expression inside the parentheses, distributing the common factor, and simplifying the expression further. The final answer is the simplified expression itself.

Tips and Tricks

• When simplifying an expression, always look for common factors that can be factored out.
• Use the distributive property to distribute the common factor to the terms inside the parentheses.
• Simplify the expression inside the parentheses before distributing the common factor.
• Check your work by plugging in values for the variable to ensure that the simplified expression is equivalent to the original expression.

Common Mistakes to Avoid

• Don't forget to factor out the greatest common factor (GCF) from the terms.
• Don't simplify the expression inside the parentheses before distributing the common factor.
• Don't forget to check your work by plugging in values for the variable to ensure that the simplified expression is equivalent to the original expression.

Real-World Applications

Simplifying expressions is an essential skill in algebra that has many real-world applications. For example, in physics, simplifying expressions is used to describe the motion of objects, while in engineering, simplifying expressions is used to design and optimize systems. In finance, simplifying expressions is used to calculate interest rates and investment returns.

Practice Problems

• Simplify the expression: $4x^2 + 2x - 5$
• Simplify the expression: $2x^2 + 3x - 1$
• Simplify the expression: $x^2 + 2x - 3$

Answer Key

• $4x^2 + 2x - 5 = 2(2x^2 + x) - 5$
• $2x^2 + 3x - 1 = (2x + 1)(x - 1)$
• $x^2 + 2x - 3 = (x + 3)(x - 1)$
  Simplify the Expression: 9x^2 + 6x - 8 - Q&A =====================================================

Q: What is the greatest common factor (GCF) of 9x^2, 6x, and -8?

A: The greatest common factor (GCF) of 9x^2, 6x, and -8 is 1, since there is no common factor that divides all three terms.

Q: Can we factor out a common factor from the first two terms?

A: Yes, we can factor out a common factor of 3 from the first two terms, which are 9x^2 and 6x. This gives us:

$$9x^2 + 6x - 8 = 3(3x^2 + 2x) - 8$$

Q: What is the expression inside the parentheses?

A: The expression inside the parentheses is 3x^2 + 2x, which can be rewritten as:

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

Q: How do we simplify the expression inside the parentheses?

A: We can simplify the expression inside the parentheses by factoring out the common factor of x, which gives us:

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

Q: What is the final simplified expression?

A: After simplifying the expression, we find that the final answer is:

$$9x^2 + 6x - 8 = 3x(3x + 2) - 8$$

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

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

• Not factoring out the greatest common factor (GCF) from the terms.
• Simplifying the expression inside the parentheses before distributing the common factor.
• Not checking your work by plugging in values for the variable to ensure that the simplified expression is equivalent to the original expression.

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

A: Simplifying expressions has many real-world applications, including:

• Physics: Simplifying expressions is used to describe the motion of objects.
• Engineering: Simplifying expressions is used to design and optimize systems.
• Finance: Simplifying expressions is used to calculate interest rates and investment returns.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through practice problems, such as:

• Simplify the expression: 4x^2 + 2x - 5
• Simplify the expression: 2x^2 + 3x - 1
• Simplify the expression: x^2 + 2x - 3

Q: What are some tips and tricks for simplifying expressions?

A: Some tips and tricks for simplifying expressions include:

• Always look for common factors that can be factored out.
• Use the distributive property to distribute the common factor to the terms inside the parentheses.
• Simplify the expression inside the parentheses before distributing the common factor.
• Check your work by plugging in values for the variable to ensure that the simplified expression is equivalent to the original expression.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions, but it's always a good idea to check your work by plugging in values for the variable to ensure that the simplified expression is equivalent to the original expression.

Q: How can I apply simplifying expressions to real-world problems?

A: You can apply simplifying expressions to real-world problems by using the techniques and strategies learned in this article. For example, in physics, you can use simplifying expressions to describe the motion of objects, while in engineering, you can use simplifying expressions to design and optimize systems. In finance, you can use simplifying expressions to calculate interest rates and investment returns.