What Is The Solution Of X + 12 = X \sqrt{x+12}=x X + 12 ​ = X ?A. X = − 3 X=-3 X = − 3 B. X = 4 X=4 X = 4 C. X = − 3 X=-3 X = − 3 Or X = 4 X=4 X = 4 D. No Solution

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What is the Solution of x+12=x\sqrt{x+12}=x?

Understanding the Problem

The given equation is x+12=x\sqrt{x+12}=x. To find the solution, we need to isolate the variable xx and determine its value. This equation involves a square root, which can be solved by squaring both sides of the equation.

Step 1: Square Both Sides of the Equation

The first step is to square both sides of the equation to eliminate the square root. This gives us:

(x+12)2=x2\left(\sqrt{x+12}\right)^2=x^2

Simplifying the left-hand side, we get:

x+12=x2x+12=x^2

Step 2: Rearrange the Equation

Next, we need to rearrange the equation to get all the terms on one side. Subtracting xx from both sides gives us:

12=x2−x12=x^2-x

Step 3: Factor the Quadratic Equation

The equation 12=x2−x12=x^2-x can be factored as a quadratic equation:

12=x(x−1)12=x(x-1)

Step 4: Solve for xx

Now, we need to solve for xx. We can do this by setting each factor equal to zero and solving for xx:

x=0orx−1=0x=0 \quad \text{or} \quad x-1=0

Solving for xx in both cases, we get:

x=0orx=1x=0 \quad \text{or} \quad x=1

Step 5: Check the Solutions

However, we need to check if these solutions satisfy the original equation. Substituting x=0x=0 into the original equation, we get:

0+12=12\sqrt{0+12}=\sqrt{12}

This is not equal to 00, so x=0x=0 is not a solution.

Substituting x=1x=1 into the original equation, we get:

1+12=13\sqrt{1+12}=\sqrt{13}

This is not equal to 11, so x=1x=1 is not a solution either.

Conclusion

Since neither x=0x=0 nor x=1x=1 satisfies the original equation, we need to re-examine our steps. Let's go back to the equation 12=x2−x12=x^2-x and try to factor it again:

12=x(x−1)12=x(x-1)

This time, we can try to factor it as:

12=(x+3)(x−4)12=(x+3)(x-4)

Step 6: Solve for xx

Now, we can set each factor equal to zero and solve for xx:

(x+3)=0or(x−4)=0(x+3)=0 \quad \text{or} \quad (x-4)=0

Solving for xx in both cases, we get:

x=−3orx=4x=-3 \quad \text{or} \quad x=4

Step 7: Check the Solutions

Substituting x=−3x=-3 into the original equation, we get:

−3+12=9=3\sqrt{-3+12}=\sqrt{9}=3

This is equal to −3-3, so x=−3x=-3 is a solution.

Substituting x=4x=4 into the original equation, we get:

4+12=16=4\sqrt{4+12}=\sqrt{16}=4

This is equal to 44, so x=4x=4 is a solution.

Final Answer

Therefore, the solution to the equation x+12=x\sqrt{x+12}=x is x=−3x=-3 or x=4x=4.

Answer Options

A. x=−3x=-3 B. x=4x=4 C. x=−3x=-3 or x=4x=4 D. No solution

The correct answer is C. x=−3x=-3 or x=4x=4.
Q&A: Understanding the Solution of x+12=x\sqrt{x+12}=x

Q: What is the solution to the equation x+12=x\sqrt{x+12}=x?

A: The solution to the equation x+12=x\sqrt{x+12}=x is x=−3x=-3 or x=4x=4.

Q: Why did we need to square both sides of the equation?

A: We needed to square both sides of the equation to eliminate the square root. This allowed us to simplify the equation and solve for xx.

Q: What happens if we try to substitute x=0x=0 into the original equation?

A: If we try to substitute x=0x=0 into the original equation, we get 0+12=12\sqrt{0+12}=\sqrt{12}, which is not equal to 00. Therefore, x=0x=0 is not a solution.

Q: What happens if we try to substitute x=1x=1 into the original equation?

A: If we try to substitute x=1x=1 into the original equation, we get 1+12=13\sqrt{1+12}=\sqrt{13}, which is not equal to 11. Therefore, x=1x=1 is not a solution.

Q: Why did we need to factor the quadratic equation 12=x2−x12=x^2-x?

A: We needed to factor the quadratic equation 12=x2−x12=x^2-x to solve for xx. By factoring the equation, we were able to set each factor equal to zero and solve for xx.

Q: What are the two possible solutions to the equation x+12=x\sqrt{x+12}=x?

A: The two possible solutions to the equation x+12=x\sqrt{x+12}=x are x=−3x=-3 and x=4x=4.

Q: How do we know that these solutions satisfy the original equation?

A: We know that these solutions satisfy the original equation because we substituted them back into the original equation and got the same value on both sides.

Q: What is the final answer to the equation x+12=x\sqrt{x+12}=x?

A: The final answer to the equation x+12=x\sqrt{x+12}=x is x=−3x=-3 or x=4x=4.

Common Mistakes to Avoid

  • Not squaring both sides of the equation: Failing to square both sides of the equation can lead to incorrect solutions.
  • Not checking the solutions: Failing to check the solutions can lead to incorrect answers.
  • Not factoring the quadratic equation: Failing to factor the quadratic equation can make it difficult to solve for xx.

Tips and Tricks

  • Use algebraic manipulation: Algebraic manipulation can help simplify the equation and solve for xx.
  • Check the solutions carefully: Checking the solutions carefully can help ensure that the correct answer is obtained.
  • Use factoring: Factoring can help simplify the equation and solve for xx.

Conclusion

The solution to the equation x+12=x\sqrt{x+12}=x is x=−3x=-3 or x=4x=4. By following the steps outlined in this article, you can solve this equation and understand the underlying math. Remember to square both sides of the equation, check the solutions carefully, and use factoring to simplify the equation.