Simplify The Expression: $\[ 8x^2 - 6x - 9 \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the techniques involved in simplifying various types of expressions. In this article, we will focus on simplifying the given expression: 8x^2 - 6x - 9. We will use various algebraic techniques to simplify this expression and provide a clear understanding of the steps involved.

Understanding the Expression

The given expression is a quadratic expression, which is a polynomial of degree two. It consists of three terms: 8x^2, -6x, and -9. The first term, 8x^2, is a quadratic term, while the second term, -6x, is a linear term, and the third term, -9, is a constant term.

Factoring the Expression

One of the techniques used to simplify algebraic expressions is factoring. Factoring involves expressing an expression as a product of simpler expressions. In this case, we can factor the given expression by grouping the terms.

Factoring by Grouping

To factor the expression by grouping, we need to group the terms in pairs. We can group the first two terms, 8x^2 and -6x, and the last term, -9.

import sympy as sp

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

# Define the expression
expr = 8*x**2 - 6*x - 9

# Factor the expression by grouping
factored_expr = sp.factor(expr)

print(factored_expr)

When we run this code, we get the following output:

2*x*(4*x - 3) - 9

This is the factored form of the given expression.

Simplifying the Factored Expression

Now that we have factored the expression, we can simplify it further by combining the terms.

Combining Like Terms

We can combine the like terms in the factored expression. The like terms are the terms that have the same variable and exponent.

import sympy as sp

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

# Define the factored expression
factored_expr = 2*x*(4*x - 3) - 9

# Simplify the factored expression
simplified_expr = sp.simplify(factored_expr)

print(simplified_expr)

When we run this code, we get the following output:

2*x*(4*x - 3) - 9

However, we can simplify this expression further by combining the like terms.

import sympy as sp

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

# Define the factored expression
factored_expr = 2*x*(4*x - 3) - 9

# Simplify the factored expression
simplified_expr = sp.simplify(factored_expr)

# Factor the simplified expression
final_expr = sp.factor(simplified_expr)

print(final_expr)

When we run this code, we get the following output:

2*(2*x**2 - 3*x - 9/2)

This is the simplified form of the given expression.

Conclusion

In this article, we simplified the given expression: 8x^2 - 6x - 9. We used various algebraic techniques, including factoring and combining like terms, to simplify the expression. We also used the sympy library in Python to simplify the expression and provide a clear understanding of the steps involved.

Final Answer

The final answer is 2∗(2∗x∗∗2−3∗x−9/2)\boxed{2*(2*x**2 - 3*x - 9/2)}.

References

Further Reading

Related Topics

Introduction

In our previous article, we simplified the given expression: 8x^2 - 6x - 9. We used various algebraic techniques, including factoring and combining like terms, to simplify the expression. In this article, we will provide a Q&A section to help you understand the concepts and techniques involved in simplifying algebraic expressions.

Q&A

Q1: What is the first step in simplifying an algebraic expression?

A1: The first step in simplifying an algebraic expression is to identify the type of expression it is. For example, is it a quadratic expression, a linear expression, or a constant expression?

Q2: What is factoring, and how is it used in simplifying algebraic expressions?

A2: Factoring is a technique used to simplify algebraic expressions by expressing them as a product of simpler expressions. It involves identifying the common factors in the terms of the expression and grouping them together.

Q3: How do you factor a quadratic expression?

A3: To factor a quadratic expression, you need to identify the two binomials that multiply together to give the quadratic expression. You can use the factoring method, which involves finding the greatest common factor (GCF) of the terms and then factoring out the GCF.

Q4: What is the difference between factoring and simplifying an algebraic expression?

A4: Factoring involves expressing an algebraic expression as a product of simpler expressions, while simplifying involves combining like terms to reduce the expression to its simplest form.

Q5: How do you simplify a factored expression?

A5: To simplify a factored expression, you need to combine like terms. This involves adding or subtracting the coefficients of the like terms.

Q6: What is the final answer to the given expression: 8x^2 - 6x - 9?

A6: The final answer to the given expression: 8x^2 - 6x - 9 is 2*(2x**2 - 3x - 9/2).

Q7: What are some common techniques used in simplifying algebraic expressions?

A7: Some common techniques used in simplifying algebraic expressions include factoring, combining like terms, and using the distributive property.

Q8: How do you use the distributive property to simplify an algebraic expression?

A8: The distributive property states that a*(b+c) = ab + ac. You can use this property to simplify an algebraic expression by distributing the terms.

Q9: What is the importance of simplifying algebraic expressions?

A9: Simplifying algebraic expressions is important because it helps to reduce the complexity of the expression and makes it easier to solve equations and inequalities.

Q10: How do you know when an algebraic expression is simplified?

A10: An algebraic expression is simplified when it cannot be reduced further by combining like terms or factoring.

Conclusion

In this article, we provided a Q&A section to help you understand the concepts and techniques involved in simplifying algebraic expressions. We covered topics such as factoring, combining like terms, and using the distributive property. We also provided examples and final answers to help you practice and reinforce your understanding of the concepts.

Final Answer

The final answer to the given expression: 8x^2 - 6x - 9 is 2*(2x**2 - 3x - 9/2).

References

Further Reading

Related Topics