Solve The Equation:${ N^2 - 6n - 93 = -7 }$

Mar 13, 2025 by ADMIN 45 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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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: $n^2 - 6n - 93 = -7$ . We will break down the solution into manageable steps, using algebraic techniques to isolate the variable $n$ .

Understanding Quadratic Equations

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, and $x$ is the variable. In our case, the equation is $n^2 - 6n - 93 = -7$ , which can be rewritten as $n^2 - 6n - 100 = 0$ by adding $7$ to both sides.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$ , the solutions are given by:

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

In our case, $a = 1$ , $b = -6$ , and $c = -100$ . Plugging these values into the formula, we get:

n = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-100)}}{2(1)}

Simplifying the Expression

Let's simplify the expression under the square root:

(-6)^2 - 4(1)(-100) = 36 + 400 = 436

So, the expression becomes:

n = \frac{6 \pm \sqrt{436}}{2}

Finding the Solutions

Now, we need to find the two solutions for $n$ . We can do this by evaluating the expression for both the plus and minus signs:

n_1 = \frac{6 + \sqrt{436}}{2}

n_2 = \frac{6 - \sqrt{436}}{2}

Evaluating the Solutions

To evaluate the solutions, we need to calculate the value of $\sqrt{436}$ . Using a calculator or a computer algebra system, we get:

\sqrt{436} \approx 20.88

Now, we can plug this value back into the expressions for $n_1$ and $n_2$ :

n_1 \approx \frac{6 + 20.88}{2} \approx 13.44

n_2 \approx \frac{6 - 20.88}{2} \approx -7.44

Conclusion

In this article, we solved the quadratic equation $n^2 - 6n - 93 = -7$ using the quadratic formula. We broke down the solution into manageable steps, simplifying the expression under the square root and evaluating the solutions. The two solutions for $n$ are approximately $13.44$ and $-7.44$ . This demonstrates the power of the quadratic formula in solving quadratic equations.

Real-World Applications

Quadratic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. Similarly, the motion of a pendulum can be described by a quadratic equation.

Tips and Tricks

When solving quadratic equations, it's essential to follow these tips and tricks:

Check your work: Always check your work by plugging the solutions back into the original equation.
Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. Use it whenever possible.
Simplify the expression: Simplify the expression under the square root to make it easier to evaluate the solutions.
Evaluate the solutions: Evaluate the solutions using a calculator or a computer algebra system.

By following these tips and tricks, you'll become proficient in solving quadratic equations and be able to tackle more complex problems in mathematics and other fields.

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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 answer some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you understand and solve these equations.

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 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: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

Factoring: If the equation can be factored into the product of two binomials, you can solve it by setting each factor equal to zero.
Quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$ , the solutions are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$ , the solutions are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, simplify the expression under the square root and evaluate the solutions.

Q: What is the difference between the two solutions?

A: The two solutions of a quadratic equation are given by the quadratic formula. The difference between the two solutions is given by: $\Delta x = \frac{\sqrt{b^2 - 4ac}}{a}$

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula gives two distinct solutions, and there is no way to have more than two solutions.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This occurs when the expression under the square root is negative, i.e., $b^2 - 4ac < 0$ .

Q: How do I check my work?

A: To check your work, plug the solutions back into the original equation. If the solutions satisfy the equation, then your work is correct.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. Similarly, the motion of a pendulum can be described by a quadratic equation.

Conclusion

In this article, we answered some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you understand and solve these equations. Whether you're a student or a professional, quadratic equations are an essential part of mathematics, and mastering them will help you tackle more complex problems in various fields.

Additional Resources

For more information on quadratic equations, check out the following resources:

Mathway: A online math problem solver that can help you solve quadratic equations.
Khan Academy: A free online resource that provides video lessons and practice exercises on quadratic equations.
Wolfram Alpha: A powerful online calculator that can help you solve quadratic equations and other mathematical problems.

By following these resources and practicing regularly, you'll become proficient in solving quadratic equations and be able to tackle more complex problems in mathematics and other fields.