Question

Factorise: 25x² − 1 − 2y − y²

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Answer 1

Given:-

• 25x² - 1 - 2y - y²

We have to factorise it.

Answer:-

Given that,

25x² - 1 - 2y - y²

= 25x² - (1 + 2y + y²)

= 25x² - (y² + 2y + 1)

Splitting the middle term,

= 25x² - (y² + y + y + 1)

= 25x² - {y(y + 1) + 1(y + 1)}

= 25x² - {(y + 1)(y + 1)}

= 25x² - (y + 1)²

= (5x)² - (y + 1)²

• Now we will use a² - b² = (a + b)(a - b) with a = 5x and b = y + 1

= {5x + (y + 1)}{5x - (y + 1)}

= (5x + y + 1)(5x - y - 1) Ans

Extra Knowledge:-

• (a + b)² = a² + b² + 2ab
• (a - b)² = a² + b² - 2ab
• a² - b² = (a + b)(a - b)
• (a + b)³ = a³ + b³ + 3ab(a + b)
• (a - b)³ = a³ - b³ - 3ab(a - b)
• a³ + b³ = (a + b)(a² + b² - ab)
• a³ - b³ = (a - b)(a² + b² + ab)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Answer 2

Question :

Factorise: 25x² − 1 − 2y − y² .

Solution :

→ 25x² - (1 + 2y + y²)

→ 25x² - (y² + 2y + 1)

Splitting the middle term,

→ 25x² - (y² + y + y + 1)

→ 25x² - {y(y + 1) + 1(y + 1)}

→ 25x² - {(y + 1)(y + 1)}

→ 25x² - (y + 1)²

→ (5x)² - (y + 1)²

Now we will use a² - b² = (a + b)(a - b) with a = 5x and b = y + 1

➡ {5x + (y + 1)} {5x - (y + 1)}

➡ (5x + y + 1) (5x - y - 1)

Additional Information :

• (a + b)² = a² + b² + 2ab
• (a - b)² = a² + b² - 2ab
• a² - b² = (a + b)(a - b)
• (a + b)³ = a³ + b³ + 3ab(a + b)
• (a - b)³ = a³ - b³ - 3ab(a - b)
• a³ + b³ = (a + b)(a² + b² - ab)
• a³ - b³ = (a - b)(a² + b² + ab)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Thank you.