Factorise
` x(x-a) - y ( y-a) `

Answer

To factorise the expression \( x(x-a) - y(y-a) \), we will follow these steps:

### Step 1: Expand the expression
Start by expanding both terms in the expression:

\[ x(x-a) = x^2 - ax \]
\[ y(y-a) = y^2 - ay \]

Thus, the expression becomes:

\[ x^2 - ax - (y^2 - ay) \]

### Step 2: Simplify the expression
Now, simplify the expression by distributing the negative sign:

\[ x^2 - ax - y^2 + ay \]

Rearranging the terms gives us:

\[ x^2 - y^2 - ax + ay \]

### Step 3: Group the terms
Next, we can group the terms in a way that makes it easier to factor:

\[ (x^2 - y^2) + (ay - ax) \]

### Step 4: Factor out common factors
Now, we can factor out the common factors from each group:

1. The first group \( x^2 - y^2 \) is a difference of squares, which can be factored as:

\[ (x - y)(x + y) \]

2. The second group \( ay - ax \) can be factored as:

\[ a(y - x) \]

Thus, we rewrite the expression as:

\[ (x - y)(x + y) + a(y - x) \]

### Step 5: Factor out the common binomial
Notice that \( y - x \) is the negative of \( x - y \). Therefore, we can rewrite \( a(y - x) \) as \( -a(x - y) \):

\[ (x - y)(x + y) - a(x - y) \]

Now, we can factor out \( (x - y) \):

\[ (x - y)((x + y) - a) \]

### Final Answer
Thus, the factorised form of the expression \( x(x-a) - y(y-a) \) is:

\[ (x - y)(x + y - a) \]

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