-(x-2)(3-2x)%3D0.jpeg "(x^2-4)-(x-2)(3-2x)=0")
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.