Solve For { X $} . . . { 2x - 16 = X(2 - X) \}

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Introduction

In this article, we will delve into the world of algebra and solve a linear equation involving a quadratic expression. The given equation is $2x - 16 = x(2 - x)$, and our goal is to isolate the variable $x$ and find its value. We will use various algebraic techniques, including expansion, simplification, and factoring, to solve for $x$.

Step 1: Expand the Right-Hand Side of the Equation

The first step in solving the equation is to expand the right-hand side, which is a quadratic expression. We can do this by multiplying the terms inside the parentheses.

$$x(2 - x) = 2x - x^2$$

Now, the equation becomes:

$$2x - 16 = 2x - x^2$$

Step 2: Simplify the Equation

Next, we can simplify the equation by combining like terms. We can subtract $2x$ from both sides of the equation to get:

$$-16 = -x^2$$

Step 3: Multiply Both Sides by -1

To make the equation more manageable, we can multiply both sides by $-1$. This will change the sign of the equation, but it will also allow us to work with a more familiar form.

$$16 = x^2$$

Step 4: Take the Square Root of Both Sides

Now, we can take the square root of both sides of the equation. This will give us two possible solutions for $x$.

$$\pm 4 = x$$

Step 5: Check the Solutions

Before we can conclude that we have found the solution to the equation, we need to check our solutions. We can do this by plugging each solution back into the original equation and verifying that it is true.

For $x = 4$, we have:

$$2(4) - 16 = 4(2 - 4)$$

$$8 - 16 = 4(-2)$$

$$-8 = -8$$

This is true, so $x = 4$ is a solution to the equation.

For $x = -4$, we have:

$$2(-4) - 16 = -4(2 - (-4))$$

$$-8 - 16 = -4(2 + 4)$$

$$-24 = -4(6)$$

$$-24 = -24$$

This is also true, so $x = -4$ is a solution to the equation.

Conclusion

In this article, we solved the equation $2x - 16 = x(2 - x)$ by expanding the right-hand side, simplifying the equation, multiplying both sides by $-1$, taking the square root of both sides, and checking the solutions. We found that the solutions to the equation are $x = 4$ and $x = -4$. These solutions satisfy the original equation and are therefore valid.

Final Answer

The final answer is $\boxed{4, -4}$.

Additional Tips and Tricks

• When solving equations involving quadratic expressions, it's often helpful to expand the right-hand side first.
• Be careful when multiplying both sides of an equation by a negative number, as this can change the sign of the equation.
• Always check your solutions by plugging them back into the original equation.

Related Topics

• Solving linear equations
• Expanding and simplifying quadratic expressions
• Factoring quadratic expressions
• Solving systems of equations

Further Reading

• For more information on solving linear equations, see our article on Solving Linear Equations.
• For more information on expanding and simplifying quadratic expressions, see our article on Expanding and Simplifying Quadratic Expressions.
• For more information on factoring quadratic expressions, see our article on Factoring Quadratic Expressions.
• For more information on solving systems of equations, see our article on Solving Systems of Equations.
  

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Introduction

In our previous article, we solved the equation $2x - 16 = x(2 - x)$ and found that the solutions to the equation are $x = 4$ and $x = -4$. In this article, we will answer some frequently asked questions about solving the equation and provide additional tips and tricks for solving similar equations.

Q&A

Q: What is the first step in solving the equation $2x - 16 = x(2 - x)$?

A: The first step in solving the equation is to expand the right-hand side, which is a quadratic expression. We can do this by multiplying the terms inside the parentheses.

Q: Why do we need to multiply both sides of the equation by $-1$?

A: We multiply both sides of the equation by $-1$ to make the equation more manageable. This will change the sign of the equation, but it will also allow us to work with a more familiar form.

Q: How do we check the solutions to the equation?

A: We can check the solutions to the equation by plugging each solution back into the original equation and verifying that it is true.

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

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

• Not expanding the right-hand side of the equation
• Not simplifying the equation
• Not checking the solutions
• Not considering the possibility of multiple solutions

Q: How do we know which solution to choose when there are multiple solutions?

A: When there are multiple solutions, we can choose the solution that is most convenient or relevant to the problem. In some cases, we may need to consider the context of the problem and choose the solution that is most appropriate.

Q: Can we use other methods to solve the equation $2x - 16 = x(2 - x)$?

A: Yes, we can use other methods to solve the equation, such as factoring or using the quadratic formula. However, these methods may be more complicated or require additional steps.

Additional Tips and Tricks

• When solving equations involving quadratic expressions, it's often helpful to expand the right-hand side first.
• Be careful when multiplying both sides of an equation by a negative number, as this can change the sign of the equation.
• Always check your solutions by plugging them back into the original equation.
• Consider using a graphing calculator or other tool to visualize the solutions to the equation.
• Be aware of the possibility of multiple solutions and consider the context of the problem when choosing a solution.

Related Topics

• Solving linear equations
• Expanding and simplifying quadratic expressions
• Factoring quadratic expressions
• Solving systems of equations

Further Reading

• For more information on solving linear equations, see our article on Solving Linear Equations.
• For more information on expanding and simplifying quadratic expressions, see our article on Expanding and Simplifying Quadratic Expressions.
• For more information on factoring quadratic expressions, see our article on Factoring Quadratic Expressions.
• For more information on solving systems of equations, see our article on Solving Systems of Equations.

Conclusion

In this article, we answered some frequently asked questions about solving the equation $2x - 16 = x(2 - x)$ and provided additional tips and tricks for solving similar equations. We hope that this article has been helpful in clarifying the steps involved in solving the equation and has provided you with a better understanding of how to approach similar problems.