4x 2 (x−1) 2 −16=0

To solve the equation \( 4x^2 (x - 1)^2 - 16 = 0 \), we can follow these steps:

1. Set the equation to zero: \[ 4x^2 (x - 1)^2 - 16 = 0 \]

2. Add 16 to both sides: \[ 4x^2 (x - 1)^2 = 16 \]

3. Divide both sides by 4: \[ x^2 (x - 1)^2 = 4 \]

4. Take the square root of both sides (consider both the positive and negative square roots): \[ x (x - 1) = 2 \quad \text{or} \quad x (x - 1) = -2 \]

Solving \( x (x - 1) = 2 \)

1. Rearranging: \[ x^2 - x - 2 = 0 \]

2. Factoring: \[ (x - 2)(x + 1) = 0 \]

3. Finding the roots: \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

Solving \( x (x - 1) = -2 \)

1. Rearranging: \[ x^2 - x + 2 = 0 \]

2. Using the quadratic formula: The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -1, c = 2 \): \[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \] \[ x = \frac{1 \pm \sqrt{1 - 8}}{2} \] \[ x = \frac{1 \pm \sqrt{-7}}{2} \] \[ x = \frac{1 \pm i\sqrt{7}}{2} \]

Summary of Solutions

The solutions to the original equation \( 4x^2 (x - 1)^2 - 16 = 0 \) are:

• \( x = 2 \)
• \( x = -1 \)
• \( x = \frac{1 + i\sqrt{7}}{2} \)
• \( x = \frac{1 - i\sqrt{7}}{2} \)

These include two real solutions (2 and -1) and two complex solutions.