What Are The Solutions Of $x^2 + 6x - 6 = 10$?A. $x = -11$ Or \$x = 1$[/tex\] B. $x = -11$ Or $x = -1$ C. \$x = -8$[/tex\] Or $x = -2$ D. $x = -8$ Or

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Introduction

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 will focus on solving the quadratic equation $x^2 + 6x - 6 = 10$ and explore the different solutions that can be obtained.

Understanding the Quadratic Equation

The given quadratic equation is in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = 6$, and $c = -16$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Rearranging the Equation

The first step in solving the quadratic equation is to rearrange it in the standard form of $ax^2 + bx + c = 0$. We can do this by subtracting $10$ from both sides of the equation:

x2+6x−6=10x^2 + 6x - 6 = 10

x2+6x−16=0x^2 + 6x - 16 = 0

Using the Quadratic Formula

One of the most common methods for solving quadratic equations is by using the quadratic formula. The quadratic formula is given by:

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

In this case, $a = 1$, $b = 6$, and $c = -16$. Plugging these values into the quadratic formula, we get:

x=−6±62−4(1)(−16)2(1)x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-16)}}{2(1)}

x=−6±36+642x = \frac{-6 \pm \sqrt{36 + 64}}{2}

x=−6±1002x = \frac{-6 \pm \sqrt{100}}{2}

x=−6±102x = \frac{-6 \pm 10}{2}

Finding the Solutions

Now that we have the quadratic formula, we can find the solutions by plugging in the values of $x$:

x=−6+102x = \frac{-6 + 10}{2}

x=42x = \frac{4}{2}

x=2x = 2

x=−6−102x = \frac{-6 - 10}{2}

x=−162x = \frac{-16}{2}

x=−8x = -8

Conclusion

In conclusion, the solutions to the quadratic equation $x^2 + 6x - 6 = 10$ are $x = 2$ and $x = -8$. These solutions can be obtained by using the quadratic formula and plugging in the values of $a$, $b$, and $c$.

Final Answer

The final answer is: 2,−8\boxed{2, -8}

Introduction

Solving quadratic equations can be a challenging task, especially for those who are new to the concept. In this article, we will address some of the most frequently asked questions about solving quadratic equations, providing clear and concise answers to help you better understand the process.

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 is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by the equation x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

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

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 and solve for x. You may need to use a calculator or a computer to simplify the expression and find the solutions.

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

A: The steps to solve a quadratic equation are:

  1. Write the equation in the standard form of ax^2 + bx + c = 0.
  2. Plug in the values of a, b, and c into the quadratic formula.
  3. Simplify the expression and solve for x.
  4. Check the solutions to make sure they are valid.

Q: What are the different types of solutions to a quadratic equation?

A: There are three types of solutions to a quadratic equation:

  1. Real and distinct solutions: These are solutions that are real numbers and are distinct from each other.
  2. Real and repeated solutions: These are solutions that are real numbers and are repeated.
  3. Complex solutions: These are solutions that are complex numbers.

Q: How do I determine the type of solution to a quadratic equation?

A: To determine the type of solution to a quadratic equation, you need to look at the discriminant (b^2 - 4ac). If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and repeated. If the discriminant is negative, the solutions are complex.

Q: What is the discriminant?

A: The discriminant is a value that is used to determine the type of solution to a quadratic equation. It is given by the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the discriminant to determine the type of solution?

A: To use the discriminant to determine the type of solution, you need to plug in the values of a, b, and c into the expression b^2 - 4ac. Then, simplify the expression and determine the type of solution based on the value of the discriminant.

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

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

  1. Not writing the equation in the standard form.
  2. Not plugging in the correct values of a, b, and c into the quadratic formula.
  3. Not simplifying the expression correctly.
  4. Not checking the solutions to make sure they are valid.

Conclusion

In conclusion, solving quadratic equations can be a challenging task, but with the right tools and techniques, it can be done. By understanding the quadratic formula, the steps to solve a quadratic equation, and the different types of solutions, you can become proficient in solving quadratic equations. Remember to avoid common mistakes and to check your solutions to make sure they are valid.

Final Answer

The final answer is: 2,−8\boxed{2, -8}