What Are The Solutions Of The Equation X 2 + 15 = 79 X^2 + 15 = 79 X 2 + 15 = 79 ?Enter Your Answer As The Correct Values, Separated By A Comma, Like This: 42, 53. If Your Answer Includes A Fraction, Use The / Symbol. For Example, If Your Answer Is

Mar 9, 2025 by ADMIN 249 views

What are the solutions of the equation $x^2 + 15 = 79$?

Introduction

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$ , where $a = 1$ , $b = 0$ , and $c = -15 + 79 = 64$ . To solve this equation, we need to isolate the variable $x$ and find its values.

Rearranging the Equation

The first step in solving the equation is to rearrange it in the standard form of a quadratic equation. We can do this by subtracting 79 from both sides of the equation:

$x^2 + 15 - 79 = 0$

This simplifies to:

$x^2 - 64 = 0$

Using the Quadratic Formula

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

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

In this case, we have $a = 1$ , $b = 0$ , and $c = -64$ . Plugging these values into the quadratic formula, we get:

$x = \frac{-0 \pm \sqrt{0^2 - 4(1)(-64)}}{2(1)}$

Simplifying this expression, we get:

$x = \frac{\pm \sqrt{256}}{2}$

$x = \frac{\pm 16}{2}$

Finding the Solutions

Now that we have the solutions in the form of $x = \frac{\pm 16}{2}$ , we can simplify them further. Dividing both the numerator and the denominator by 2, we get:

$x = \pm 8$

Therefore, the solutions to the equation $x^2 + 15 = 79$ are $x = 8$ and $x = -8$ .

Conclusion

In this article, we solved the quadratic equation $x^2 + 15 = 79$ using the quadratic formula. We rearranged the equation in the standard form, applied the quadratic formula, and simplified the solutions to find the correct values. The solutions to the equation are $x = 8$ and $x = -8$ .

Discussion

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods and techniques used to find the solutions. In this article, we focused on solving the equation $x^2 + 15 = 79$ and provided the correct values as the solutions. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions to a wide range of equations.

Applications

Quadratic equations have numerous applications in mathematics, science, and engineering. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits. In this article, we focused on solving a quadratic equation, but the techniques and methods used can be applied to a wide range of problems.

Future Work

In the future, we can explore other methods for solving quadratic equations, such as factoring and completing the square. We can also apply the quadratic formula to more complex equations and explore the properties of quadratic functions.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Quadratic Formula" by Khan Academy
[3] "Solving Quadratic Equations" by Purplemath

Acknowledgments

I would like to thank my instructor and peers for their guidance and support throughout this project. I would also like to thank the authors of the references cited in this article for their contributions to the field of mathematics.

Introduction

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 (usually x) is two. It is typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

Q: What are the different methods for solving quadratic equations?

A: There are several methods for solving quadratic equations, including:

Factoring: This involves expressing the quadratic equation as a product of two binomials.
Quadratic Formula: This involves using the quadratic formula to find the solutions.
Completing the Square: This involves rewriting the quadratic equation in a form that allows us to easily find the solutions.
Graphing: This involves graphing the quadratic equation on a coordinate plane to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows us to find the solutions to a quadratic equation. It is given by:

x = (-b ± √(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 from the quadratic equation into the formula. Then, simplify the expression 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:

Plug in the values of a, b, and c into the quadratic formula.
Simplify the expression.
Solve for x.

Q: What are the advantages and disadvantages of using the quadratic formula?

A: The advantages of using the quadratic formula are:

It is a powerful tool for solving quadratic equations.
It can be used to find the solutions to a wide range of quadratic equations.
It is a simple and straightforward method.

The disadvantages of using the quadratic formula are:

It can be difficult to use if the quadratic equation is complex.
It can be time-consuming to simplify the expression.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex solutions. However, you need to be careful when simplifying the expression, as complex numbers can be tricky to work with.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula are:

Not plugging in the correct values of a, b, and c.
Not simplifying the expression correctly.
Not solving for x correctly.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with rational solutions. However, you need to be careful when simplifying the expression, as rational numbers can be tricky to work with.

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

A: Quadratic equations have numerous real-world applications, including:

Modeling the motion of objects.
Finding the maximum or minimum value of a function.
Solving problems in physics, engineering, and economics.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about solving quadratic equations. We discussed the different methods for solving quadratic equations, including the quadratic formula, and provided tips and tricks for using the quadratic formula. We also discussed some common mistakes to avoid and some real-world applications of quadratic equations.