Find And Correct The Error In The Following Equation:$\[ \begin{aligned} 2x^2 - 2x - 24 & = 2\left(x^2 - X - 12\right) \\ & = 2(x - 3)(x + 4) \end{aligned} \\] The Error Is In The Factorization Step. The Correct Factorization Of \[$x^2 - X

Understanding the Given Equation

The given equation is a quadratic equation in the form of $2x^2 - 2x - 24 = 2(x - 3)(x + 4)$. However, the error lies in the factorization step. To identify the mistake, we need to carefully analyze the equation and understand the correct factorization of the quadratic expression.

Correct Factorization of Quadratic Expression

The quadratic expression $x^2 - x - 12$ can be factored into $(x - 4)(x + 3)$. This is the correct factorization of the quadratic expression.

Step-by-Step Solution

To find the error in the given equation, we need to follow the step-by-step solution:

Step 1: Factorize the Quadratic Expression

The quadratic expression $x^2 - x - 12$ can be factored into $(x - 4)(x + 3)$.

Step 2: Identify the Error

The error in the given equation lies in the factorization step. The correct factorization of the quadratic expression is $(x - 4)(x + 3)$, not $(x - 3)(x + 4)$.

Step 3: Correct the Error

To correct the error, we need to replace the incorrect factorization with the correct one. The corrected equation is:

$$2x^2 - 2x - 24 = 2(x - 4)(x + 3)$$

Solving the Corrected Equation

Now that we have corrected the error, we can solve the equation:

$$2x^2 - 2x - 24 = 2(x - 4)(x + 3)$$

Step 1: Expand the Right-Hand Side

We can expand the right-hand side of the equation using the distributive property:

$$2x^2 - 2x - 24 = 2(x^2 + 3x - 4x - 12)$$

Step 2: Simplify the Right-Hand Side

We can simplify the right-hand side of the equation by combining like terms:

$$2x^2 - 2x - 24 = 2(x^2 - x - 12)$$

Step 3: Factorize the Right-Hand Side

We can factorize the right-hand side of the equation using the correct factorization:

$$2x^2 - 2x - 24 = 2(x - 4)(x + 3)$$

Step 4: Solve for x

We can solve for x by setting the left-hand side equal to the right-hand side:

$$2x^2 - 2x - 24 = 2(x - 4)(x + 3)$$

We can expand the right-hand side and simplify the equation:

$$2x^2 - 2x - 24 = 2x^2 - 8x + 6x - 24$$

We can combine like terms:

$$2x^2 - 2x - 24 = 2x^2 - 2x - 24$$

This is a true statement, which means that the equation is an identity.

Conclusion

In conclusion, the error in the given equation lies in the factorization step. The correct factorization of the quadratic expression is $(x - 4)(x + 3)$, not $(x - 3)(x + 4)$. We have corrected the error and solved the equation to find that it is an identity.

Key Takeaways

• The correct factorization of the quadratic expression $x^2 - x - 12$ is $(x - 4)(x + 3)$.
• The error in the given equation lies in the factorization step.
• We have corrected the error and solved the equation to find that it is an identity.

Final Answer

The final answer is: $\boxed{0}$

Understanding the Given Equation

The given equation is a quadratic equation in the form of $2x^2 - 2x - 24 = 2(x - 3)(x + 4)$. However, the error lies in the factorization step. To identify the mistake, we need to carefully analyze the equation and understand the correct factorization of the quadratic expression.

Q&A Session

Q: What is the correct factorization of the quadratic expression $x^2 - x - 12$?

A: The correct factorization of the quadratic expression $x^2 - x - 12$ is $(x - 4)(x + 3)$.

Q: Where does the error lie in the given equation?

A: The error in the given equation lies in the factorization step. The correct factorization of the quadratic expression is $(x - 4)(x + 3)$, not $(x - 3)(x + 4)$.

Q: How do we correct the error in the given equation?

A: To correct the error, we need to replace the incorrect factorization with the correct one. The corrected equation is:

$$2x^2 - 2x - 24 = 2(x - 4)(x + 3)$$

Q: How do we solve the corrected equation?

A: To solve the corrected equation, we need to follow the step-by-step solution:

1. Expand the right-hand side of the equation using the distributive property.
2. Simplify the right-hand side of the equation by combining like terms.
3. Factorize the right-hand side of the equation using the correct factorization.
4. Solve for x by setting the left-hand side equal to the right-hand side.

Q: What is the final answer to the corrected equation?

A: The final answer to the corrected equation is 0, which means that the equation is an identity.

Common Mistakes in Algebraic Equations

Mistake 1: Incorrect Factorization

One common mistake in algebraic equations is incorrect factorization. This can lead to incorrect solutions and a deeper understanding of the problem.

Mistake 2: Not Checking the Work

Another common mistake in algebraic equations is not checking the work. This can lead to incorrect solutions and a deeper understanding of the problem.

Mistake 3: Not Using the Correct Method

A third common mistake in algebraic equations is not using the correct method. This can lead to incorrect solutions and a deeper understanding of the problem.

Conclusion

In conclusion, the error in the given equation lies in the factorization step. The correct factorization of the quadratic expression is $(x - 4)(x + 3)$, not $(x - 3)(x + 4)$. We have corrected the error and solved the equation to find that it is an identity.

Key Takeaways

• The correct factorization of the quadratic expression $x^2 - x - 12$ is $(x - 4)(x + 3)$.
• The error in the given equation lies in the factorization step.
• We have corrected the error and solved the equation to find that it is an identity.

Final Answer

The final answer is: $\boxed{0}$

Additional Resources

For more information on algebraic equations and how to solve them, please refer to the following resources:

• Algebraic Equations
• Solving Quadratic Equations
• Factorization

Practice Problems

To practice solving algebraic equations, please try the following problems:

• Solve the equation $x^2 + 5x + 6 = 0$.
• Solve the equation $x^2 - 7x + 12 = 0$.
• Solve the equation $x^2 + 2x - 15 = 0$.

Conclusion

In conclusion, the error in the given equation lies in the factorization step. The correct factorization of the quadratic expression is $(x - 4)(x + 3)$, not $(x - 3)(x + 4)$. We have corrected the error and solved the equation to find that it is an identity.