Solve For $x$ In The Equation $x^2 + 4x - 4 = 8$.A. \$x = -6$[/tex\] Or $x = 2$B. $x = -2 \pm 2\sqrt{2}$C. \$x = -2$[/tex\] Or $x = 6$D. $x = 2 \pm 2\sqrt{2}$

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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, $x^2 + 4x - 4 = 8$, and explore the different methods and techniques used 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. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and completing the square.

The Given Equation

The given equation is $x^2 + 4x - 4 = 8$. To solve for $x$, we need to isolate the variable on one side of the equation. The first step is to move all the terms to one side of the equation by subtracting 8 from both sides:

x2+4x−4−8=0x^2 + 4x - 4 - 8 = 0

This simplifies to:

x2+4x−12=0x^2 + 4x - 12 = 0

Using 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, $a = 1$, $b = 4$, and $c = -12$. Plugging these values into the quadratic formula, we get:

x=−4±42−4(1)(−12)2(1)x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-12)}}{2(1)}

Simplifying the expression under the square root, we get:

x=−4±16+482x = \frac{-4 \pm \sqrt{16 + 48}}{2}

x=−4±642x = \frac{-4 \pm \sqrt{64}}{2}

x=−4±82x = \frac{-4 \pm 8}{2}

This gives us two possible solutions:

x=−4+82=2x = \frac{-4 + 8}{2} = 2

x=−4−82=−6x = \frac{-4 - 8}{2} = -6

Alternative Methods

In addition to the quadratic formula, there are other methods for solving quadratic equations, such as factoring and completing the square. However, these methods may not be as straightforward or efficient as the quadratic formula.

Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. In our case, the equation $x^2 + 4x - 12 = 0$ can be factored as:

(x+6)(x−2)=0(x + 6)(x - 2) = 0

This gives us two possible solutions:

x+6=0⇒x=−6x + 6 = 0 \Rightarrow x = -6

x−2=0⇒x=2x - 2 = 0 \Rightarrow x = 2

Completing the Square

Completing the square involves rewriting the quadratic equation in a form that allows us to easily identify the solutions. In our case, the equation $x^2 + 4x - 12 = 0$ can be rewritten as:

(x+2)2−16=0(x + 2)^2 - 16 = 0

This gives us two possible solutions:

(x+2)2=16⇒x+2=±4(x + 2)^2 = 16 \Rightarrow x + 2 = \pm 4

x=−2±4x = -2 \pm 4

This gives us two possible solutions:

x=−2+4=2x = -2 + 4 = 2

x=−2−4=−6x = -2 - 4 = -6

Conclusion

In this article, we have solved the quadratic equation $x^2 + 4x - 4 = 8$ using the quadratic formula, factoring, and completing the square. We have shown that the solutions to the equation are $x = -6$ or $x = 2$. These solutions can be verified by plugging them back into the original equation.

Answer

The correct answer is:

A. $x = -6$ or $x = 2$

This answer is consistent with the solutions obtained using the quadratic formula, factoring, and completing the square.

Discussion

The solution to the quadratic equation $x^2 + 4x - 4 = 8$ is a classic example of how to use the quadratic formula, factoring, and completing the square to solve quadratic equations. These methods are essential tools for students and professionals alike, and are used extensively in mathematics, science, and engineering.

Additional Resources

For those who want to learn more about quadratic equations and how to solve them, there are many online resources available, including:

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

Frequently Asked Questions

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

  • Factoring: Expressing the quadratic equation as a product of two binomials.
  • Quadratic Formula: Using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.
  • Completing the Square: Rewriting the quadratic equation in a form that allows us to easily identify the solutions.

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 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. For example, if you have the equation $x^2 + 4x - 12 = 0$, you would plug in $a = 1$, $b = 4$, and $c = -12$ into the formula.

Q: What is the difference between factoring and completing the square?

A: Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves rewriting the quadratic equation in a form that allows us to easily identify the solutions.

Q: Can I use the quadratic formula to solve all quadratic equations?

A: Yes, the quadratic formula can be used to solve all quadratic equations. However, it may not always be the most efficient or easiest method to use.

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

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

  • Not checking the solutions to see if they are valid.
  • Not using the correct method for solving the equation.
  • Not simplifying the solutions to their simplest form.

Q: How do I check if my solutions are valid?

A: To check if your solutions are valid, you need to plug them back into the original equation and see if they satisfy the equation.

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

A: Quadratic equations have many 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.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the different methods for solving quadratic equations, including factoring, the quadratic formula, and completing the square, you can become proficient in solving these equations and apply them to real-world problems.

Additional Resources

For those who want to learn more about quadratic equations and how to solve them, there are many online resources available, including:

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

These resources provide a wealth of information and practice problems to help you master the art of solving quadratic equations.