What Is The Solution To $2x^2 + 8x = X^2 - 16$?A. $x = -4$ B. $x = -2$ C. $x = 2$ D. $x = 4$

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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 delve into the solution of a quadratic equation, specifically the equation $2x^2 + 8x = x^2 - 16$. We will break down the steps to solve this equation and provide the final solution.

Understanding the Equation

The given equation is a quadratic equation, which is a polynomial equation of degree two. It can be written in the general form as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this case, the equation can be rewritten as $2x^2 + 8x - x^2 + 16 = 0$.

Rearranging the Equation

To solve the equation, we need to rearrange it in the standard form of a quadratic equation. We can do this by combining like terms:

2x2+8x−x2+16=02x^2 + 8x - x^2 + 16 = 0

⇒2x2−x2+8x+16=0\Rightarrow 2x^2 - x^2 + 8x + 16 = 0

⇒x2+8x+16=0\Rightarrow x^2 + 8x + 16 = 0

Solving the Equation

Now that we have the equation in the standard form, we can solve it 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, $a = 1$, $b = 8$, and $c = 16$. Plugging these values into the quadratic formula, we get:

x=−8±82−4(1)(16)2(1)x = \frac{-8 \pm \sqrt{8^2 - 4(1)(16)}}{2(1)}

⇒x=−8±64−642\Rightarrow x = \frac{-8 \pm \sqrt{64 - 64}}{2}

⇒x=−8±02\Rightarrow x = \frac{-8 \pm \sqrt{0}}{2}

⇒x=−82\Rightarrow x = \frac{-8}{2}

⇒x=−4\Rightarrow x = -4

Conclusion

Therefore, the solution to the quadratic equation $2x^2 + 8x = x^2 - 16$ is $x = -4$. This is the only solution to the equation, and it can be verified by plugging it back into the original equation.

Final Answer

The final answer is: −4\boxed{-4}

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 frequently asked questions (FAQs) about quadratic equations, including their definition, types, and solutions.

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 (usually x) is two. It can be written in the general form as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What are the Types of Quadratic Equations?

A: There are three types of quadratic equations:

  • Monic Quadratic Equation: A quadratic equation of the form $x^2 + bx + c = 0$, where the coefficient of the $x^2$ term is 1.
  • Non-Monic Quadratic Equation: A quadratic equation of the form $ax^2 + bx + c = 0$, where the coefficient of the $x^2$ term is not 1.
  • Perfect Square Quadratic Equation: A quadratic equation that can be factored as a perfect square, such as $(x + a)^2 = 0$.

Q: How Do I Solve a Quadratic Equation?

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

  • Factoring: If the quadratic equation can be factored, you can solve it by finding the factors.
  • Quadratic Formula: If the quadratic equation cannot be factored, you can use the quadratic formula to solve it.
  • Graphing: You can also solve a quadratic equation by graphing it on a coordinate plane.

Q: What is the Quadratic Formula?

A: The quadratic formula is a formula that can be used to solve a quadratic equation. It is given by:

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

Q: What is the Difference Between the Two Solutions of a Quadratic Equation?

A: The two solutions of a quadratic equation are given by the quadratic formula. The difference between the two solutions is given by:

Δx=b2−4aca\Delta x = \frac{\sqrt{b^2 - 4ac}}{a}

Q: Can a Quadratic Equation Have More Than Two Solutions?

A: No, a quadratic equation can only have two solutions. This is because the quadratic formula always gives two 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 discriminant ($b^2 - 4ac$) is negative.

Q: Can a Quadratic Equation Have One Solution?

A: Yes, a quadratic equation can have one solution. This occurs when the discriminant ($b^2 - 4ac$) is zero.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about quadratic equations, including their definition, types, and solutions. We hope that this article has provided you with a better understanding of quadratic equations and how to solve them.

Final Answer

The final answer is: 0\boxed{0}