Solve The Equation: N 2 + 8 N + 15 = 0 N^2 + 8n + 15 = 0 N 2 + 8 N + 15 = 0 Choose The Correct Solution From The Options Below:1. 3 2. 2 3. 1 4. 4

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation: $n^2 + 8n + 15 = 0$. We will break down the solution process into manageable steps and provide a clear explanation of each step.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $n$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

This formula will be used to solve the given quadratic equation.

Solving the Equation

Now, let's apply the quadratic formula to solve the equation $n^2 + 8n + 15 = 0$.

Step 1: Identify the Coefficients

The given equation is $n^2 + 8n + 15 = 0$. We need to identify the coefficients $a$, $b$, and $c$.

• $a = 1$ (coefficient of $n^2$)
• $b = 8$ (coefficient of $n$)
• $c = 15$ (constant term)

Step 2: Plug in the Values

Now, we will plug in the values of $a$, $b$, and $c$ into the quadratic formula.

$$n = \frac{-8 \pm \sqrt{8^2 - 4(1)(15)}}{2(1)}$$

Step 3: Simplify the Expression

Next, we will simplify the expression under the square root.

$$n = \frac{-8 \pm \sqrt{64 - 60}}{2}$$

$$n = \frac{-8 \pm \sqrt{4}}{2}$$

$$n = \frac{-8 \pm 2}{2}$$

Step 4: Solve for $n$

Now, we will solve for $n$ by considering both the positive and negative cases.

• Case 1: $n = \frac{-8 + 2}{2}$
• Case 2: $n = \frac{-8 - 2}{2}$

Case 1: $n = \frac{-8 + 2}{2}$

$$n = \frac{-6}{2}$$

$$n = -3$$

Case 2: $n = \frac{-8 - 2}{2}$

$$n = \frac{-10}{2}$$

$$n = -5$$

Conclusion

We have solved the quadratic equation $n^2 + 8n + 15 = 0$ using the quadratic formula. The solutions are $n = -3$ and $n = -5$. Therefore, the correct solution from the options below is:

• 3 is incorrect
• 2 is incorrect
• 1 is incorrect
• 4 is incorrect

The correct answer is not listed among the options. However, based on the solutions we obtained, we can see that the correct answer is not among the options. The correct answer is $n = -3$ or $n = -5$.

Discussion

This problem is a great example of how to use the quadratic formula to solve quadratic equations. The quadratic formula is a powerful tool that can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$. In this problem, we used the quadratic formula to solve the equation $n^2 + 8n + 15 = 0$. We obtained two solutions: $n = -3$ and $n = -5$. These solutions can be verified by plugging them back into the original equation.

Practice Problems

If you want to practice solving quadratic equations, try the following problems:

• $x^2 + 5x + 6 = 0$
• $y^2 - 4y + 4 = 0$
• $z^2 + 2z + 1 = 0$

Conclusion

$$$$$$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we solved the quadratic equation $n^2 + 8n + 15 = 0$ using the quadratic formula. In this article, we will provide a Q&A guide to help you understand quadratic equations and how to solve them.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $n$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

This formula will be used to solve quadratic equations of the form $ax^2 + bx + c = 0$.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to identify the coefficients $a$, $b$, and $c$ in the quadratic equation. Then, plug in the values of $a$, $b$, and $c$ into the quadratic formula. Simplify the expression under the square root and solve for $x$.

Q: What are the steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

1. Identify the coefficients $a$, $b$, and $c$ in the quadratic equation.
2. Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
3. Simplify the expression under the square root.
4. Solve for $x$.

Q: What are the two solutions to a quadratic equation?

A: The two solutions to a quadratic equation are given by the quadratic formula:

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

These two solutions are:

• $x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$
• $x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$

Q: How do I determine which solution is correct?

A: To determine which solution is correct, you need to plug both solutions back into the original equation and check if they satisfy the equation. If both solutions satisfy the equation, then both solutions are correct. If only one solution satisfies the equation, then that solution is the correct solution.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations are:

• Not identifying the coefficients $a$, $b$, and $c$ correctly.
• Not simplifying the expression under the square root correctly.
• Not solving for $x$ correctly.
• Not checking if both solutions satisfy the equation.

Conclusion

In this article, we provided a Q&A guide to help you understand quadratic equations and how to solve them. We discussed the quadratic formula, the steps to solve a quadratic equation using the quadratic formula, and some common mistakes to avoid. We hope this guide has been helpful in understanding quadratic equations and how to solve them.

Practice Problems

If you want to practice solving quadratic equations, try the following problems:

• $x^2 + 5x + 6 = 0$
• $y^2 - 4y + 4 = 0$
• $z^2 + 2z + 1 = 0$

Additional Resources

If you want to learn more about quadratic equations, we recommend the following resources:

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

We hope this guide has been helpful in understanding quadratic equations and how to solve them. If you have any questions or need further clarification, please don't hesitate to ask.