Solve for `x` if `x(4x-4)= -4`

Answer

To solve the equation \( x(4x - 4) = -4 \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation in a standard form.

\[ x(4x - 4) = -4 \]

### Step 2: Expand the left side
Distribute \( x \) on the left side:

\[ 4x^2 - 4x = -4 \]

### Step 3: Move all terms to one side
Add 4 to both sides to set the equation to zero:

\[ 4x^2 - 4x + 4 = 0 \]

### Step 4: Identify coefficients
Now, identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c = 0 \):

- \( a = 4 \)
- \( b = -4 \)
- \( c = 4 \)

### Step 5: Use the quadratic formula
Now, apply the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot 4}}{2 \cdot 4} \]

### Step 6: Simplify the expression
Calculate the discriminant:

\[ x = \frac{4 \pm \sqrt{16 - 64}}{8} \]

\[ x = \frac{4 \pm \sqrt{-48}}{8} \]

### Step 7: Simplify the square root of a negative number
Since the discriminant is negative, we can express it using imaginary numbers:

\[ \sqrt{-48} = \sqrt{48} \cdot i = 4\sqrt{3} \cdot i \]

### Step 8: Substitute back into the equation
Now substitute back into the equation:

\[ x = \frac{4 \pm 4\sqrt{3}i}{8} \]

### Step 9: Simplify the fraction
This simplifies to:

\[ x = \frac{1 \pm \sqrt{3}i}{2} \]

### Final Answer
Thus, the solutions for \( x \) are:

\[ x = \frac{1 + \sqrt{3}i}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{3}i}{2} \]

---