Question

factorise: 2ab minus a square minus b square +1

Answer 1

Answer:

( 1- a + b) ( 1+ a- b)

Step-by-step explanation:

Used identity

Answer 2

Answer:

2ab - a² - b² + 1 = (1 - a + b)(1 + a - b)

Step-by-step explanation:

We can solve it in many ways,

Method 1

We have,

2ab - a² - b² + 1

Taking common factor (-1).

= (-1)(-2ab + a² + b² - 1)

= (-1)(a² - 2ab + b² - 1²)

Using the identity,

a² - 2ab + b² = (a - b)²

= (-1)((a - b)² - 1²)

Using the identity,

x² - y² = (x + y)(x - y)

= (-1)((a - b + 1)(a - b - 1))

= (-1)(a - b + 1)(a - b - 1)

Rearranging

= (-1)(a - b - 1)(a - b + 1)

= (-a + b + 1)(a - b + 1)

Rearranging,

= (1 - a + b)(1 + a - b)

Method 2

We have,

2ab - a² - b² + 1

Rearranging,

1 - a² + 2ab - b²

Taking the common factor out,

1 - (a² - 2ab + b²)

Using the identity,

a² - 2ab + b² = (a - b)²

So,

1 - (a² - 2ab + b²)

= 1 - (a - b)²

We can say,

1 = 1²

So,

= 1² - (a - b)²

Using the identity,

x² - y² = (x + y)(x - y)

Thus,

= (1 + a - b)(1 - (a - b))

= (1 + a - b)(1 - a + b)

Rearranging,

= (1 - a + b)(1 + a - b)

Hence,

2ab - a² - b² + 1 = (1 - a + b)(1 + a - b)

Hope it helped and believing you understood it. All the best.