Solve the following for x using the square root property.

sup(bracket(4x-1),2)=-396

Enter one value per answer box. If there are more answer boxes than solutions, leave the extra boxes blank. Make sure to fully simplify any fractional coefficients, and enter complex numbers in standard form.

x=

To solve the equation \( (4x - 1)^2 = -396 \) using the square root property, follow these steps:

### Step 1: Isolate the squared term
The equation is already in the form \( (4x - 1)^2 = -396 \).

### Step 2: Take the square root of both sides
Taking the square root of both sides, we get:
\[
4x - 1 = \pm \sqrt{-396}
\]

### Step 3: Simplify the square root
Since \( \sqrt{-396} \) involves a negative number under the square root, it will yield a complex number:
\[
\sqrt{-396} = \sqrt{396} \cdot \sqrt{-1} = \sqrt{396}i
\]
Next, simplify \( \sqrt{396} \). Since \( 396 = 4 \times 99 \), we have:
\[
\sqrt{396} = \sqrt{4 \times 99} = 2\sqrt{99}
\]
Therefore:
\[
4x - 1 = \pm 2\sqrt{99}i
\]

### Step 4: Solve for x
Add 1 to both sides:
\[
4x = 1 \pm 2\sqrt{99}i
\]
Now, divide both sides by 4:
\[
x = \frac{1 \pm 2\sqrt{99}i}{4}
\]
This simplifies to:
\[
x = \frac{1}{4} \pm \frac{\sqrt{99}i}{2}
\]

### Final Answer
The solutions for \( x \) are:
\[
x = \frac{1}{4} + \frac{\sqrt{99}i}{2}, \quad x = \frac{1}{4} - \frac{\sqrt{99}i}{2}
\]

You can enter these values in the answer boxes.

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Would you like more details or have any questions? Here are some related problems to consider:

1. Solve \( (3x + 2)^2 = -100 \) for \( x \).
2. Solve \( (5x - 7)^2 = -225 \) using the square root property.
3. Find the complex roots of \( x^2 + 16 = 0 \).
4. Solve \( (2x - 3)^2 = 49 \) for \( x \).
5. What is the square root of \( -64 \) in standard form?

**Tip:** Remember, the square root of a negative number always introduces an imaginary unit \( i \) into the solution.

Learn how to solve a quadratic equation involving complex numbers using the square root property. This detailed solution covers algebraic concepts such as simplifying square roots, dealing with imaginary units, and solving quadratic equations. Perfect for students in Grades 9-11.

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