Evaluate The Expression: − 1.5 ± ( 1.5 ) 2 − 4 ( − 192 ) ( 1.5 ) 2 ( − 192 ) \frac{-1.5 \pm \sqrt{(1.5)^2-4(-192)(1.5)}}{2(-192)} 2 ( − 192 ) − 1.5 ± ( 1.5 ) 2 − 4 ( − 192 ) ( 1.5 ) ​ ​

Introduction

In mathematics, evaluating expressions is a crucial aspect of problem-solving, and it requires a deep understanding of various mathematical concepts. The given expression, $\frac{-1.5 \pm \sqrt{(1.5)^2-4(-192)(1.5)}}{2(-192)}$, is a complex equation that involves square roots, fractions, and variables. In this article, we will delve into the world of mathematics and provide a step-by-step analysis of the given expression.

Understanding the Expression

The given expression is a quadratic equation in the form of $\frac{-1.5 \pm \sqrt{a^2-4bc}}{2c}$. To evaluate this expression, we need to understand the concept of quadratic equations and how to simplify them. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2+bx+c=0$, where $a$, $b$, and $c$ are constants.

Breaking Down the Expression

To evaluate the given expression, we need to break it down into smaller components. The expression can be rewritten as $\frac{-1.5 \pm \sqrt{(1.5)^2-4(-192)(1.5)}}{2(-192)}$. Let's start by simplifying the expression inside the square root.

Simplifying the Expression Inside the Square Root

The expression inside the square root is $(1.5)^2-4(-192)(1.5)$. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent: $(1.5)^2 = 2.25$
2. Multiply $-192$ and $1.5$: $-192 \times 1.5 = -288$
3. Subtract $-288$ from $2.25$: $2.25 - (-288) = 2.25 + 288 = 290.25$

Simplifying the Expression

Now that we have simplified the expression inside the square root, we can rewrite the original expression as $\frac{-1.5 \pm \sqrt{290.25}}{2(-192)}$. The next step is to simplify the square root.

Simplifying the Square Root

The square root of $290.25$ is approximately $17.05$. Therefore, the expression can be rewritten as $\frac{-1.5 \pm 17.05}{2(-192)}$.

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it. The expression can be rewritten as $\frac{-1.5 + 17.05}{2(-192)}$ or $\frac{-1.5 - 17.05}{2(-192)}$. Let's evaluate both expressions.

Evaluating the First Expression

The first expression is $\frac{-1.5 + 17.05}{2(-192)}$. To evaluate this expression, we need to follow the order of operations:

1. Add $-1.5$ and $17.05$: $-1.5 + 17.05 = 15.55$
2. Divide $15.55$ by $2(-192)$: $\frac{15.55}{-384} = -0.0405$

Evaluating the Second Expression

The second expression is $\frac{-1.5 - 17.05}{2(-192)}$. To evaluate this expression, we need to follow the order of operations:

1. Subtract $-1.5$ and $17.05$: $-1.5 - 17.05 = -18.55$
2. Divide $-18.55$ by $2(-192)$: $\frac{-18.55}{-384} = 0.0484$

Conclusion

In conclusion, the given expression $\frac{-1.5 \pm \sqrt{(1.5)^2-4(-192)(1.5)}}{2(-192)}$ can be evaluated by simplifying the expression inside the square root and then evaluating the resulting expression. The final answer is a range of values, which is $-0.0405$ and $0.0484$. This range of values is a result of the $\pm$ symbol in the original expression.

Final Answer

The final answer is $\boxed{-0.0405, 0.0484}$.

References

• [1] Khan Academy. (n.d.). Quadratic Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations
• [2] Mathway. (n.d.). Quadratic Equation Solver. Retrieved from https://www.mathway.com/subjects/quadratic-equations

Related Topics

• Quadratic Equations
• Square Roots
• Fractions
• Variables

Tags

• Quadratic Equations
• Square Roots
• Fractions
• Variables
• Math
• Algebra
• Mathematics
  

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Introduction

In our previous article, we evaluated the expression $\frac{-1.5 \pm \sqrt{(1.5)^2-4(-192)(1.5)}}{2(-192)}$. In this article, we will answer some frequently asked questions related to the evaluation of this expression.

Q: What is the significance of the $\pm$ symbol in the expression?

A: The $\pm$ symbol in the expression indicates that there are two possible values for the expression. The $\pm$ symbol is used to represent the two possible values of the expression, which are $-0.0405$ and $0.0484$.

Q: How do I simplify the expression inside the square root?

A: To simplify the expression inside the square root, you need to follow the order of operations (PEMDAS). First, evaluate the exponent: $(1.5)^2 = 2.25$. Then, multiply $-192$ and $1.5$: $-192 \times 1.5 = -288$. Finally, subtract $-288$ from $2.25$: $2.25 - (-288) = 2.25 + 288 = 290.25$.

Q: What is the significance of the square root in the expression?

A: The square root in the expression is used to find the two possible values of the expression. The square root of $290.25$ is approximately $17.05$. This value is then used to find the two possible values of the expression, which are $-0.0405$ and $0.0484$.

Q: How do I evaluate the expression?

A: To evaluate the expression, you need to follow the order of operations. First, simplify the expression inside the square root. Then, evaluate the square root. Finally, divide the result by $2(-192)$.

Q: What is the final answer to the expression?

A: The final answer to the expression is a range of values, which is $-0.0405$ and $0.0484$.

Q: Can I use a calculator to evaluate the expression?

A: Yes, you can use a calculator to evaluate the expression. However, it's always a good idea to understand the steps involved in evaluating the expression.

Q: What are some common mistakes to avoid when evaluating the expression?

A: Some common mistakes to avoid when evaluating the expression include:

• Not following the order of operations
• Not simplifying the expression inside the square root
• Not evaluating the square root correctly
• Not dividing the result by $2(-192)$

Q: How do I apply the concept of evaluating expressions to real-world problems?

A: The concept of evaluating expressions can be applied to real-world problems in various fields, such as science, engineering, and finance. For example, in science, you may need to evaluate expressions to model the behavior of a physical system. In engineering, you may need to evaluate expressions to design a system or a structure. In finance, you may need to evaluate expressions to model the behavior of financial markets.

Q: What are some additional resources that I can use to learn more about evaluating expressions?

A: Some additional resources that you can use to learn more about evaluating expressions include:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Conclusion

In conclusion, evaluating the expression $\frac{-1.5 \pm \sqrt{(1.5)^2-4(-192)(1.5)}}{2(-192)}$ requires a deep understanding of various mathematical concepts, including quadratic equations, square roots, and fractions. By following the steps outlined in this article, you can evaluate the expression and find the two possible values, which are $-0.0405$ and $0.0484$.