i. Let’s write for which value of t, will be a whole square form.

ii. Let’s write the expression which when added to gives a whole square.

iii. If a and b are positive integers and lets write the value of a & b.

iv. an identity or an equation? Write with reason.

v. For each positive or negative value of x and y except zero the value of is always (positive or negative)

i. For making a complete square, we need to reduce the expression in x²± 2xy + y²form, so if we put t = ±1, then we get expression as

Hence, for t = ±1, the given equation will be a whole square form.

ii. For making a complete square, we need to reduce the expression in x² + 2xy + y²form, so if we add ±4x to given equation, we have

a² + 4 ± 4x

= a²± 4x + 4

= a²± 2(2)x + 2²

= (a ± 2)²

Hence, if we add 4x to the given expression, the expression becomes a whole square.

iii. Given,

a² – b² = 9 × 11

we know, a² – b² = (a - b)(a + b)

⇒ (a – b)(a + b) = 9 × 11

On comparison, we get

a – b = 9 [1]

a + b = 11 [2]

Adding equation [1] and [2]

a + b + a – b = 9 + 11

⇒ 2a = 20

⇒ a = 10

Putting value of ‘a’ in equation [1], we get

10 – b = 9

⇒ b = 10 – 9

⇒ b = 1

iv. Identity, as

Taking LHS

Now, we know

(x + y)² = x² + 2xy + y²

and (x – y)² = x² – 2xy + y²

Applying above identities in LHS, we get

= (x + y)² – (x – y)²

= x² + 2xy + y² – (x² – 2xy + y²)

= x² + 2xy + y² – x² + 2xy – y²

= 4xy

= RHS

As, LHS = RHS the given expression is an identity!

v. Square of any number is always positive

[Explanation: Suppose -a (a > 0) is a negative number, then

(-a)² = -a × (-a) = a²]

Therefore, a²and b² both are positive numbers

⇒ a² + b² is a positive number, as sum of two positive numbers is positive!