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

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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 the quadratic equation $x^2 + 15 = 79$, and we will provide a step-by-step guide on how to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants. In our case, the equation is $x^2 + 15 = 79$, which can be rewritten as $x^2 - 64 = 0$ by subtracting 79 from 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=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

x=−0±02−4(1)(−64)2(1)x = \frac{-0 \pm \sqrt{0^2 - 4(1)(-64)}}{2(1)}

Simplifying the Quadratic Formula

Simplifying the expression under the square root, we get:

x=±2562x = \frac{\pm \sqrt{256}}{2}

x=±162x = \frac{\pm 16}{2}

Finding the Solutions

Now that we have simplified the quadratic formula, we can find the solutions by plugging in the values of x. We get:

x=162=8x = \frac{16}{2} = 8

x=−162=−8x = \frac{-16}{2} = -8

Conclusion

In this article, we have solved the quadratic equation $x^2 + 15 = 79$ using the quadratic formula. We have shown that the solutions are x = 8 and x = -8. This demonstrates the power of the quadratic formula in solving quadratic equations.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Tips and Tricks

Here are some tips and tricks for solving quadratic equations:

  • Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
  • Simplify the expression: Simplify the expression under the square root to make it easier to solve.
  • Check your solutions: Check your solutions to make sure they are correct.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic equations:

  • Not using the quadratic formula: Failing to use the quadratic formula can lead to incorrect solutions.
  • Not simplifying the expression: Failing to simplify the expression under the square root can make it difficult to solve.
  • Not checking solutions: Failing to check solutions can lead to incorrect answers.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By using the quadratic formula and simplifying the expression, we can find the solutions to quadratic equations. Remember to check your solutions and avoid common mistakes to ensure accurate results.

Final Answer

The final answer is: 8,-8

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Introduction

Quadratic equations can be a challenging topic for many students and professionals. In this article, we will answer some of the most frequently asked questions about quadratic equations, including how to solve them, common mistakes to avoid, and real-world applications.

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, x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Alternatively, you can try to factor the equation or use the method of completing the square.

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=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I simplify the quadratic formula?

A: To simplify the quadratic formula, you can start by simplifying the expression under the square root. This will make it easier to solve the equation.

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

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

  • Not using the quadratic formula: Failing to use the quadratic formula can lead to incorrect solutions.
  • Not simplifying the expression: Failing to simplify the expression under the square root can make it difficult to solve.
  • Not checking solutions: Failing to check solutions can lead to incorrect answers.

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

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

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Q: How do I check my solutions?

A: To check your solutions, you can plug the values of x back into the original equation and see if they are true. You can also use a calculator or computer program to check your solutions.

Q: What are some tips and tricks for solving quadratic equations?

A: Some tips and tricks for solving quadratic equations include:

  • Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
  • Simplify the expression: Simplify the expression under the square root to make it easier to solve.
  • Check your solutions: Check your solutions to make sure they are correct.

Q: Can I use a calculator or computer program to solve quadratic equations?

A: Yes, you can use a calculator or computer program to solve quadratic equations. Many calculators and computer programs have built-in functions for solving quadratic equations.

Conclusion

In conclusion, quadratic equations can be a challenging topic, but with the right tools and techniques, you can solve them with ease. Remember to use the quadratic formula, simplify the expression, and check your solutions to ensure accurate results.

Final Answer

The final answer is: There is no one-size-fits-all answer to this question, as the solution to a quadratic equation depends on the specific equation and the values of a, b, and c. However, by following the steps outlined in this article, you can solve quadratic equations with confidence.