Factorise the following fully: 20x² - 45y²

Answer

To factorise the expression 20x² - 45y², we will follow these steps: 1. **Identify the common factors**: We need to observe if there are any common numerical factors in the coefficients of the terms. In this case, the coefficients are 20 and 45. The greatest common divisor (GCD) of 20 and 45 is 5. 2. **Factor out the GCD**: We can factor out 5 from the expression: \[ 20x² - 45y² = 5(4x² - 9y²) \] 3. **Recognize the difference of squares**: Now, we notice that the expression inside the parentheses, 4x² - 9y², is a difference of squares. Recall that the difference of squares can be expressed as: \[ a² - b² = (a - b)(a + b) \] In this case, \( a = 2x \) and \( b = 3y \). 4. **Apply the difference of squares formula**: Now we can apply the formula to factor 4x² - 9y²: \[ 4x² - 9y² = (2x - 3y)(2x + 3y) \] 5. **Combine the factors**: Now we can replace the original expression with our factors: \[ 20x² - 45y² = 5(2x - 3y)(2x + 3y) \] Therefore, the fully factorised form of the expression 20x² - 45y² is: \[ 5(2x - 3y)(2x + 3y) \]