Factorise : `((25)/(4)x^2-(1)/(9)y^2)`

Answer

To factorise the expression \(\frac{25}{4}x^2 - \frac{1}{9}y^2\), we can follow these steps:

### Step 1: Rewrite the expression in terms of perfect squares
We can express \(\frac{25}{4}x^2\) and \(\frac{1}{9}y^2\) as perfect squares:
\[ \frac{25}{4}x^2 = \left(\frac{5x}{2}\right)^2 \]
\[ \frac{1}{9}y^2 = \left(\frac{y}{3}\right)^2 \]

### Step 2: Substitute the perfect squares into the expression
Now we can rewrite the original expression using these perfect squares:
\[ \left(\frac{5x}{2}\right)^2 - \left(\frac{y}{3}\right)^2 \]

### Step 3: Apply the difference of squares formula
The expression is now in the form of \(A^2 - B^2\), where \(A = \frac{5x}{2}\) and \(B = \frac{y}{3}\). We can use the difference of squares formula:
\[ A^2 - B^2 = (A + B)(A - B) \]

### Step 4: Substitute \(A\) and \(B\) into the formula
Using the values of \(A\) and \(B\):
\[ \left(\frac{5x}{2} + \frac{y}{3}\right)\left(\frac{5x}{2} - \frac{y}{3}\right) \]

### Step 5: Final factorised form
Thus, the factorised form of the expression \(\frac{25}{4}x^2 - \frac{1}{9}y^2\) is:
\[ \left(\frac{5x}{2} + \frac{y}{3}\right)\left(\frac{5x}{2} - \frac{y}{3}\right) \]