The Equation $2r^2 + 1r - 4 = 0$ Has Solutions Of The Form $r = \frac{N \pm \sqrt{D}}{M}$.(A) Solve This Equation And Record The Values Of $N, M$, And $D$. Do Not Worry About Simplifying The $\sqrt{D}$

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Introduction

In this article, we will be solving a quadratic equation of the form $2r^2 + 1r - 4 = 0$. This equation is a classic example of a quadratic equation, and it can be solved using the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations, and it is widely used in mathematics and other fields.

The Quadratic Formula

The quadratic formula is given by:

$$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. In our case, the equation is $2r^2 + 1r - 4 = 0$, so we have:

$$a = 2, b = 1, c = -4$$

Substituting the Values into the Quadratic Formula

Now that we have the values of $a$, $b$, and $c$, we can substitute them into the quadratic formula:

$$r = \frac{-1 \pm \sqrt{1^2 - 4(2)(-4)}}{2(2)}$$

Simplifying the Expression

Now, let's simplify the expression under the square root:

$$1^2 - 4(2)(-4) = 1 + 32 = 33$$

So, the expression becomes:

$$r = \frac{-1 \pm \sqrt{33}}{4}$$

Recording the Values of $N$, $M$, and $D$

We are asked to record the values of $N$, $M$, and $D$. From the solution, we can see that:

$$N = -1, M = 4, D = 33$$

Therefore, the values of $N$, $M$, and $D$ are $-1$, $4$, and $33$, respectively.

Conclusion

In this article, we solved the quadratic equation $2r^2 + 1r - 4 = 0$ using the quadratic formula. We found that the solutions have the form $r = \frac{-1 \pm \sqrt{33}}{4}$. We also recorded the values of $N$, $M$, and $D$, which are $-1$, $4$, and $33$, respectively.

The Importance of Quadratic Equations

Quadratic equations are an important part of mathematics, and they have many real-world applications. They are used in a wide range of fields, including physics, engineering, and economics. Quadratic equations can be used to model real-world situations, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.

The Quadratic Formula: A Powerful Tool

The quadratic formula is a powerful tool for solving quadratic equations. It is a simple and elegant formula that can be used to solve a wide range of quadratic equations. The quadratic formula is widely used in mathematics and other fields, and it is an essential tool for anyone who wants to solve quadratic equations.

Solving Quadratic Equations: A Step-by-Step Guide

Solving quadratic equations can be a challenging task, but it can be made easier by following a step-by-step guide. Here are the steps to solve a quadratic equation:

1. Write down the quadratic equation in the form $ar^2 + br + c = 0$.
2. Identify the values of $a$, $b$, and $c$.
3. Substitute the values of $a$, $b$, and $c$ into the quadratic formula.
4. Simplify the expression under the square root.
5. Write down the solutions in the form $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Real-World Applications of Quadratic Equations

Quadratic equations have many real-world applications. Here are a few examples:

• Physics: Quadratic equations are used to model the motion of objects. For example, the trajectory of a projectile can be modeled using a quadratic equation.
• Engineering: Quadratic equations are used to design electrical circuits. For example, the behavior of a resistor-capacitor circuit can be modeled using a quadratic equation.
• Economics: Quadratic equations are used to model the behavior of economic systems. For example, the growth of a population can be modeled using a quadratic equation.

Conclusion

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Introduction

Quadratic equations are a fundamental part of mathematics, and they have many real-world applications. In our previous article, we solved the quadratic equation $2r^2 + 1r - 4 = 0$ using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two. It is typically written in the form $ar^2 + br + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

$$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, you substitute these values into the quadratic formula and simplify the expression under the square root.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions.

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when the quadratic equation cannot be factored easily. This is often the case when the equation has complex solutions or when the solutions are not rational numbers.

Q: Can I use the quadratic formula to solve cubic equations?

A: No, the quadratic formula can only be used to solve quadratic equations. Cubic equations require a different formula, known as Cardano's formula.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$. The discriminant determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: Can I use the quadratic formula to solve equations with complex coefficients?

A: Yes, the quadratic formula can be used to solve equations with complex coefficients. However, the solutions will be complex numbers.

Q: How do I simplify the expression under the square root in the quadratic formula?

A: To simplify the expression under the square root, you need to factor the expression and then take the square root of each factor.

Q: Can I use the quadratic formula to solve equations with rational coefficients?

A: Yes, the quadratic formula can be used to solve equations with rational coefficients. However, the solutions may not be rational numbers.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations. We hope that this article has provided you with a better understanding of quadratic equations and how to use the quadratic formula to solve them. If you have any further questions, please don't hesitate to ask.

Additional Resources

If you want to learn more about quadratic equations and the quadratic formula, here are some additional resources that you may find helpful:

• Textbooks: There are many textbooks that cover quadratic equations and the quadratic formula. Some popular textbooks include "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart.
• Online Resources: There are many online resources that provide information and examples of quadratic equations and the quadratic formula. Some popular online resources include Khan Academy, Mathway, and Wolfram Alpha.
• Practice Problems: Practice problems are an essential part of learning quadratic equations and the quadratic formula. You can find practice problems in textbooks, online resources, and practice problem books.

Conclusion

In conclusion, quadratic equations are an important part of mathematics, and they have many real-world applications. The quadratic formula is a powerful tool for solving quadratic equations, and it is an essential tool for anyone who wants to solve quadratic equations. We hope that this article has provided you with a better understanding of quadratic equations and how to use the quadratic formula to solve them. If you have any further questions, please don't hesitate to ask.