Solve for `x` if `4x(5-3x)= -2`

Answer

To solve the equation \( 4x(5 - 3x) = -2 \), we will follow these steps:

### Step 1: Expand the equation
Start by distributing \( 4x \) across the terms inside the parentheses:

\[ 4x(5 - 3x) = 20x - 12x^2 \]

So, we rewrite the equation as:

\[ 20x - 12x^2 = -2 \]

### Step 2: Rearrange the equation
Next, we move all terms to one side of the equation to set it to zero:

\[ -12x^2 + 20x + 2 = 0 \]

To make it easier to work with, we can multiply the entire equation by -1:

\[ 12x^2 - 20x - 2 = 0 \]

### Step 3: Identify coefficients
Now, we identify the coefficients \( A \), \( B \), and \( C \) for the quadratic formula \( Ax^2 + Bx + C = 0 \):

- \( A = 12 \)
- \( B = -20 \)
- \( C = -2 \)

### Step 4: Apply the quadratic formula
We will use the quadratic formula to find the values of \( x \):

\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Substituting the values of \( A \), \( B \), and \( C \):

\[ x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 12 \cdot (-2)}}{2 \cdot 12} \]

### Step 5: Simplify the expression
Calculating inside the square root:

\[ x = \frac{20 \pm \sqrt{400 + 96}}{24} \]

\[ x = \frac{20 \pm \sqrt{496}}{24} \]

### Step 6: Further simplify
Now, we simplify \( \sqrt{496} \):

\[ \sqrt{496} = \sqrt{16 \cdot 31} = 4\sqrt{31} \]

Substituting back into the equation:

\[ x = \frac{20 \pm 4\sqrt{31}}{24} \]

### Step 7: Final simplification
We can simplify this further:

\[ x = \frac{5 \pm \sqrt{31}}{6} \]

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

\[ x = \frac{5 + \sqrt{31}}{6} \quad \text{and} \quad x = \frac{5 - \sqrt{31}}{6} \]