## Solving the Equation (x^2-4)-(x-2)(3-2x)=0

This article will guide you through solving the equation (x^2-4)-(x-2)(3-2x)=0. We will use algebraic manipulation and simplification to find the solution.

### Step 1: Expand the expression

First, we need to expand the product (x-2)(3-2x) using the distributive property (or FOIL method): (x-2)(3-2x) = 3x - 2x² - 6 + 4x = -2x² + 7x - 6

Now the equation becomes: (x² - 4) - (-2x² + 7x - 6) = 0

### Step 2: Simplify the equation

Next, we simplify by removing the parentheses and combining like terms: x² - 4 + 2x² - 7x + 6 = 0 3x² - 7x + 2 = 0

### Step 3: Factor the quadratic equation

The simplified equation is a quadratic equation, which we can factor into two binomials: (3x - 1)(x - 2) = 0

### Step 4: Solve for x

For the product of two factors to be zero, at least one of them must be zero. Therefore, we have two possible solutions:

- 3x - 1 = 0 => x = 1/3
- x - 2 = 0 => x = 2

### Conclusion

The solutions to the equation (x² - 4) - (x - 2)(3 - 2x) = 0 are **x = 1/3** and **x = 2**. You can verify these solutions by substituting them back into the original equation and confirming that they make the equation true.