Select All Expressions That Could Be Equivalent To $x^2 + Bx - 36$ Where $b$ Is Negative.A. $(x + 3)(x - 12)$B. $ ( X − 2 ) ( X + 18 ) (x - 2)(x + 18) ( X − 2 ) ( X + 18 ) [/tex]C. $(x - 13)(x + 3)$D. $(x + 4)(x + 9)$E.

Introduction

In algebra, quadratic expressions are a fundamental concept that plays a crucial role in solving various mathematical problems. A quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. The general form of a quadratic expression is ${ax^2 + bx + c}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable. In this article, we will explore the equivalence of quadratic expressions, focusing on the given expression ${x^2 + bx - 36}$ where ${b}$ is negative.

Understanding the Given Expression

The given expression is ${x^2 + bx - 36}$, where ${b}$ is a negative constant. This means that the coefficient of the linear term, ${b}$, is a negative number. Our goal is to find all possible expressions that are equivalent to the given expression.

Equivalence of Quadratic Expressions

Two quadratic expressions are equivalent if they can be transformed into each other through algebraic manipulations, such as factoring, expanding, or combining like terms. In other words, two expressions are equivalent if they have the same value for all possible values of the variable.

Factoring Quadratic Expressions

One way to find equivalent expressions is to factor the given expression. Factoring involves expressing a quadratic expression as a product of two binomials. The general form of a factored quadratic expression is ${(x + p)(x + q)}$, where ${p}$ and ${q}$ are constants.

Factoring the Given Expression

To factor the given expression ${x^2 + bx - 36}$, we need to find two numbers whose product is ${-36}$ and whose sum is ${b}$. Since ${b}$ is negative, we know that one of the numbers must be positive, and the other must be negative.

Possible Factorizations

Let's consider the possible factorizations of the given expression:

• ${(x + 3)(x - 12)}$
• ${(x - 2)(x + 18)}$
• ${(x - 13)(x + 3)}$
• ${(x + 4)(x + 9)}$

Analyzing Each Option

Now, let's analyze each option to determine if it is equivalent to the given expression.

Option A: ${(x + 3)(x - 12)}$

To verify if this option is equivalent, we need to expand the expression and compare it with the given expression.

import sympy as sp
x = sp.symbols('x')
expr = (x + 3)*(x - 12)

expanded_expr = sp.expand(expr)
print(expanded_expr)

The expanded expression is ${x^2 - 9x - 36}$. Comparing this with the given expression, we see that the coefficients of the linear term are different. Therefore, option A is not equivalent to the given expression.

Option B: ${(x - 2)(x + 18)}$

Similarly, we can expand this expression and compare it with the given expression.

import sympy as sp
x = sp.symbols('x')

expr = (x - 2)*(x + 18)

expanded_expr = sp.expand(expr)
print(expanded_expr)

The expanded expression is ${x^2 + 16x - 36}$. Comparing this with the given expression, we see that the coefficients of the linear term are different. Therefore, option B is not equivalent to the given expression.

Option C: ${(x - 13)(x + 3)}$

We can expand this expression and compare it with the given expression.

import sympy as sp
x = sp.symbols('x')

expr = (x - 13)*(x + 3)

expanded_expr = sp.expand(expr)
print(expanded_expr)

The expanded expression is ${x^2 - 10x - 39}$. Comparing this with the given expression, we see that the coefficients of the linear term are different. Therefore, option C is not equivalent to the given expression.

Option D: ${(x + 4)(x + 9)}$

We can expand this expression and compare it with the given expression.

import sympy as sp
x = sp.symbols('x')

expr = (x + 4)*(x + 9)

expanded_expr = sp.expand(expr)
print(expanded_expr)

The expanded expression is ${x^2 + 13x + 36}$. Comparing this with the given expression, we see that the coefficients of the linear term are different. Therefore, option D is not equivalent to the given expression.

Conclusion

In conclusion, none of the options A, B, C, or D are equivalent to the given expression ${x^2 + bx - 36}$ where ${b}$ is negative. However, we can still find an equivalent expression by factoring the given expression.

Factoring the Given Expression

To factor the given expression, we need to find two numbers whose product is ${-36}$ and whose sum is ${b}$. Since ${b}$ is negative, we know that one of the numbers must be positive, and the other must be negative.

Finding the Equivalent Expression

After factoring the given expression, we get ${(x + p)(x + q)}$, where ${p}$ and ${q}$ are constants. To find the equivalent expression, we need to expand this factored expression and compare it with the given expression.

Expanding the Factored Expression

We can expand the factored expression ${(x + p)(x + q)}$ as follows:

import sympy as sp
x = sp.symbols('x')

expr = (x + 6)*(x - 6)

expanded_expr = sp.expand(expr)
print(expanded_expr)

The expanded expression is ${x^2 - 36}$. Comparing this with the given expression, we see that the constant term is different. However, we can still find an equivalent expression by adjusting the coefficients of the linear term.

Adjusting the Coefficients

To adjust the coefficients of the linear term, we can multiply the factored expression by a constant. Let's multiply the factored expression by ${-1}$.

import sympy as sp
x = sp.symbols('x')

expr = -1*(x + 6)*(x - 6)

expanded_expr = sp.expand(expr)
print(expanded_expr)

The expanded expression is ${-x^2 + 36}$. Comparing this with the given expression, we see that the constant term is different. However, we can still find an equivalent expression by adjusting the coefficients of the linear term.

Final Equivalent Expression

After adjusting the coefficients of the linear term, we get the final equivalent expression ${-x^2 + 36}$. This expression is equivalent to the given expression ${x^2 + bx - 36}$ where ${b}$ is negative.

Conclusion

In conclusion, we have found the final equivalent expression ${-x^2 + 36}$ for the given expression ${x^2 + bx - 36}$ where ${b}$ is negative. This expression is equivalent to the given expression, and it satisfies the condition that ${b}$ is negative.

Q: What is the general form of a quadratic expression?

A: The general form of a quadratic expression is ${ax^2 + bx + c}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable.

Q: What is the difference between equivalent and identical expressions?

A: Equivalent expressions are expressions that can be transformed into each other through algebraic manipulations, such as factoring, expanding, or combining like terms. Identical expressions are expressions that are exactly the same, with no differences in their coefficients or structure.

Q: How do you determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you need to check if they can be transformed into each other through algebraic manipulations. You can do this by factoring, expanding, or combining like terms.

Q: What is the significance of factoring quadratic expressions?

A: Factoring quadratic expressions is significant because it allows you to simplify complex expressions and identify their roots. Factoring also helps you to identify equivalent expressions and to solve equations.

Q: How do you factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. You can then use these numbers to write the expression as a product of two binomials.

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

A: A quadratic expression is a polynomial expression of degree two, which means the highest power of the variable is two. A polynomial expression can have any degree, not just two.

Q: Can a quadratic expression have a negative leading coefficient?

A: Yes, a quadratic expression can have a negative leading coefficient. In this case, the expression will have a negative value for all values of the variable.

Q: How do you determine if a quadratic expression is equivalent to a given expression?

A: To determine if a quadratic expression is equivalent to a given expression, you need to check if they can be transformed into each other through algebraic manipulations. You can do this by factoring, expanding, or combining like terms.

Q: What is the significance of equivalent expressions in algebra?

A: Equivalent expressions are significant in algebra because they allow you to simplify complex expressions and identify their roots. Equivalent expressions also help you to solve equations and to identify patterns in algebraic expressions.

Q: Can a quadratic expression have a negative constant term?

A: Yes, a quadratic expression can have a negative constant term. In this case, the expression will have a negative value for all values of the variable.

Q: How do you determine if a quadratic expression is equivalent to a given expression with a negative constant term?

A: To determine if a quadratic expression is equivalent to a given expression with a negative constant term, you need to check if they can be transformed into each other through algebraic manipulations. You can do this by factoring, expanding, or combining like terms.

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 the highest power of the variable is two. A linear expression is a polynomial expression of degree one, which means the highest power of the variable is one.

Q: Can a quadratic expression have a negative leading coefficient and a negative constant term?

A: Yes, a quadratic expression can have a negative leading coefficient and a negative constant term. In this case, the expression will have a negative value for all values of the variable.

Q: How do you determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient and a negative constant term?

A: To determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient and a negative constant term, you need to check if they can be transformed into each other through algebraic manipulations. You can do this by factoring, expanding, or combining like terms.

Q: What is the significance of equivalent expressions in solving equations?

A: Equivalent expressions are significant in solving equations because they allow you to simplify complex expressions and identify their roots. Equivalent expressions also help you to solve equations and to identify patterns in algebraic expressions.

Q: Can a quadratic expression have a negative leading coefficient, a negative constant term, and a negative coefficient of the linear term?

A: Yes, a quadratic expression can have a negative leading coefficient, a negative constant term, and a negative coefficient of the linear term. In this case, the expression will have a negative value for all values of the variable.

Q: How do you determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient, a negative constant term, and a negative coefficient of the linear term?

A: To determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient, a negative constant term, and a negative coefficient of the linear term, you need to check if they can be transformed into each other through algebraic manipulations. You can do this by factoring, expanding, or combining like terms.

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

A: A quadratic expression is a polynomial expression of degree two, which means the highest power of the variable is two. A rational expression is a fraction of two polynomial expressions.

Q: Can a quadratic expression have a negative leading coefficient, a negative constant term, and a negative coefficient of the linear term, and be a rational expression?

A: Yes, a quadratic expression can have a negative leading coefficient, a negative constant term, and a negative coefficient of the linear term, and be a rational expression. In this case, the expression will have a negative value for all values of the variable.

Q: How do you determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and is a rational expression?

A: To determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and is a rational expression, you need to check if they can be transformed into each other through algebraic manipulations. You can do this by factoring, expanding, or combining like terms.

Q: What is the significance of equivalent expressions in algebraic manipulations?

A: Equivalent expressions are significant in algebraic manipulations because they allow you to simplify complex expressions and identify their roots. Equivalent expressions also help you to solve equations and to identify patterns in algebraic expressions.

Q: Can a quadratic expression have a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and be a rational expression, and have a negative value for all values of the variable?

A: Yes, a quadratic expression can have a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and be a rational expression, and have a negative value for all values of the variable.

Q: How do you determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and is a rational expression, and has a negative value for all values of the variable?

A: To determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and is a rational expression, and has a negative value for all values of the variable, you need to check if they can be transformed into each other through algebraic manipulations. You can do this by factoring, expanding, or combining like terms.

Q: What is the difference between a quadratic expression and a polynomial expression of degree three?

A: A quadratic expression is a polynomial expression of degree two, which means the highest power of the variable is two. A polynomial expression of degree three is a polynomial expression with the highest power of the variable being three.

Q: Can a quadratic expression have a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and be a rational expression, and have a negative value for all values of the variable, and be a polynomial expression of degree three?

A: No, a quadratic expression cannot have a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and be a rational expression, and have a negative value for all values of the variable, and be a polynomial expression of degree three.

Q: How do you determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and is a rational expression, and has a negative value for all values of the variable, and is a polynomial expression of degree three?

A: To determine if a quadratic expression is equivalent to a given expression with a negative leading coefficient, a negative constant term, a negative coefficient of the linear term, and is a rational expression, and has a negative value for all values of the variable, and is a polynomial expression of degree three, you need to check if they can be transformed into each other through algebraic manipulations. You can do this by factoring, expanding, or combining like terms.

Q: What is the significance of equivalent expressions in solving equations and algebraic manipulations?

A: Equivalent expressions are significant in solving equations and algebraic manipulations because they allow you to simplify complex expressions and identify their roots. Equivalent expressions also help you to solve equations and to identify patterns in algebraic expressions.

**