Expand the following using the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca: Class 9

Expand the following using the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca: Class 9

Question 1.

Expand the following using the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca: Class 9

i. (p + 3q + 7r)²

ii. (3x - 2y + 4z)²

Solution:

Here, a = p, b = 3q, c = 7r

∴ (p + 3q + 7r)²

= p² + (3q)² + (7r)² + 2(p)(3q) + 2(3q)(7r) + 2(7r)(p)

= p² + 9q² + 49r² + 6pq + 42qr + 14pr

ii. Here, a = 3x, b = -2y, c = 4z

∴ (3x - 2y + 4z)² = (3x)² + (-2y)² + (4z)² + 2(3x)(-2y) + 2(-2y)(4z) + 2(4z)(3z)

= 9x² + 4y² + 16z² - 12xy - 16yz + 24xz

Question 2.

Is this an identity?

(a + b - c)² + (a - b + c)² + (a - b - c)² Class 9

= 2a² + 2b² + 2c²

Solution:

(a + b - c)² + (a - b + c)² + (a - b - c)²

= [a + b + (- c)]² + [a + (-b) + c]² + [a + (-b) +(- c)]²

= (a² + b² + c² + 2ab - 2bc - 2ac) + (a² + b² + c² - 2ab - 2bc + 2ac) + (a² + b² + c² - 2ab + 2bc - 2ac)

= 3a² + 3b² + 3c² + (2ab - 2ab - 2ab) + (-2bc - 2bc + 2bc) + (-2ac + 2ac - 2ac)

= 3a² + 3b² + 3c² - 2ab - 2bc - 2ac

≠ 2a² + 2b² + 2c²

Hence, the given statement is not an identity.