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solve this equation by using complete the square 

The equation provided is:

\[
(5x - 3)^2 = -3
\]

Before attempting to solve this by completing the square, it's important to note that a square of a real number can never be negative. This means that the equation has no real solutions.

However, we can solve it for complex solutions. Here's how:

1. Start with the given equation:
   \[
   (5x - 3)^2 = -3
   \]

2. Take the square root of both sides:
   \[
   5x - 3 = \pm \sqrt{-3}
   \]

   The square root of \(-3\) is \( \pm i\sqrt{3} \), where \(i\) is the imaginary unit.

   So we have:
   \[
   5x - 3 = \pm i\sqrt{3}
   \]

3. Solve for \(x\):
   \[
   5x = 3 \pm i\sqrt{3}
   \]
   \[
   x = \frac{3 \pm i\sqrt{3}}{5}
   \]

Thus, the solutions are:

\[
x = \frac{3 + i\sqrt{3}}{5} \quad \text{or} \quad x = \frac{3 - i\sqrt{3}}{5}
\]

These are the complex solutions to the given equation.

Would you like more details on any of the steps, or do you have further questions?

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Here are some related questions for practice:
1. What does it mean for a number to be imaginary?
2. How do you find the square root of a negative number?
3. Can completing the square method be applied to all quadratic equations?
4. What are some real-world applications of complex numbers?
5. How would the solution change if the right-hand side was positive?

**Tip:** When solving quadratic equations, always check if the discriminant (the part under the square root) is negative, as that will lead to complex solutions.

Learn how to solve the equation (5x - 3)^2 = -3 and find the complex solutions. This step-by-step solution involves using the square root property and understanding imaginary numbers. Suitable for students in Grades 10-12.

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