Solve $(2k+3)^2 - 24 = 3$Step-by-Step SolutionStep 1: Isolate The Expression Containing The Square Term.

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Step 1: Isolate the expression containing the square term

To solve the given equation, we first need to isolate the expression containing the square term. The equation is $(2k+3)^2 - 24 = 3.$ Our goal is to isolate the term $(2k+3)^2.$ To do this, we can add 24 to both sides of the equation.

(2k+3)^2 - 24 + 24 = 3 + 24

This simplifies to:

(2k+3)^2 = 27

Step 2: Take the square root of both sides

Now that we have isolated the square term, we can take the square root of both sides of the equation. This will give us:

\sqrt{(2k+3)^2} = \sqrt{27}

Since the square root of a squared term is equal to the absolute value of the term, we can simplify this to:

|2k+3| = \sqrt{27}

Step 3: Simplify the square root

The square root of 27 can be simplified as follows:

\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}

So, our equation becomes:

|2k+3| = 3\sqrt{3}

Step 4: Solve for k

Now that we have simplified the square root, we can solve for k. We have two possible cases:

Case 1: 2k+3 = 3\sqrt{3}

In this case, we can solve for k by subtracting 3 from both sides and then dividing by 2:

2k = 3\sqrt{3} - 3
k = \frac{3\sqrt{3} - 3}{2}

Case 2: 2k+3 = -3\sqrt{3}

In this case, we can solve for k by subtracting 3 from both sides and then dividing by 2:

2k = -3\sqrt{3} - 3
k = \frac{-3\sqrt{3} - 3}{2}

Step 5: Check the solutions

Now that we have found two possible solutions for k, we need to check if they are valid. We can do this by plugging each solution back into the original equation and checking if it is true.

Case 1: k = \frac{3\sqrt{3} - 3}{2}

Plugging this solution back into the original equation, we get:

(2\left(\frac{3\sqrt{3} - 3}{2}\right)+3)^2 - 24 = 3

Simplifying this expression, we get:

(3\sqrt{3} - 3 + 3)^2 - 24 = 3
(3\sqrt{3})^2 - 24 = 3
27 - 24 = 3
3 = 3

This shows that the solution k = \frac{3\sqrt{3} - 3}{2} is valid.

Case 2: k = \frac{-3\sqrt{3} - 3}{2}

Plugging this solution back into the original equation, we get:

(2\left(\frac{-3\sqrt{3} - 3}{2}\right)+3)^2 - 24 = 3

Simplifying this expression, we get:

(-3\sqrt{3} - 3 + 3)^2 - 24 = 3
(-3\sqrt{3})^2 - 24 = 3
27 - 24 = 3
3 = 3

This shows that the solution k = \frac{-3\sqrt{3} - 3}{2} is also valid.

Conclusion

In this article, we have solved the equation $(2k+3)^2 - 24 = 3$ step-by-step. We first isolated the expression containing the square term, then took the square root of both sides, simplified the square root, and finally solved for k. We found two possible solutions for k and checked if they are valid by plugging them back into the original equation. Both solutions were found to be valid.

Final Answer

The final answer is 33−32,−33−32\boxed{\frac{3\sqrt{3} - 3}{2}, \frac{-3\sqrt{3} - 3}{2}}.

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Frequently Asked Questions

Q: What is the first step in solving the equation $(2k+3)^2 - 24 = 3$?

A: The first step in solving the equation is to isolate the expression containing the square term. This can be done by adding 24 to both sides of the equation.

Q: How do I simplify the square root of 27?

A: The square root of 27 can be simplified as follows:

\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}

Q: What are the two possible cases for solving the equation ∣2k+3∣=33|2k+3| = 3\sqrt{3}?

A: The two possible cases are:

  • Case 1: 2k+3=332k+3 = 3\sqrt{3}
  • Case 2: 2k+3=−332k+3 = -3\sqrt{3}

Q: How do I solve for k in each of the two cases?

A: To solve for k in each of the two cases, you can follow these steps:

  • Case 1: Subtract 3 from both sides and then divide by 2:
2k = 3\sqrt{3} - 3
k = \frac{3\sqrt{3} - 3}{2}
  • Case 2: Subtract 3 from both sides and then divide by 2:
2k = -3\sqrt{3} - 3
k = \frac{-3\sqrt{3} - 3}{2}

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

A: To check if the solutions are valid, you can plug each solution back into the original equation and check if it is true.

Q: What is the final answer to the equation $(2k+3)^2 - 24 = 3$?

A: The final answer is 33−32,−33−32\boxed{\frac{3\sqrt{3} - 3}{2}, \frac{-3\sqrt{3} - 3}{2}}.

Common Mistakes to Avoid

1. Not isolating the expression containing the square term

Make sure to add 24 to both sides of the equation to isolate the expression containing the square term.

2. Not simplifying the square root

Make sure to simplify the square root of 27 as follows:

\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}

3. Not checking if the solutions are valid

Make sure to plug each solution back into the original equation and check if it is true.

Additional Resources

Conclusion

In this article, we have answered some of the most frequently asked questions about solving the equation $(2k+3)^2 - 24 = 3$. We have covered the first step in solving the equation, simplifying the square root, solving for k in each of the two cases, and checking if the solutions are valid. We have also provided some common mistakes to avoid and additional resources for further learning.