Question

(x + y)2 = x2 + y2 + 2xy

prove the given identity

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Answer 1

ANSWER:- (x + y)² = x² + 2xy + y²

The square of the sum of two quantities is equal to the square of the first term, plus twice the product of the two terms, plus the square of the second term.

We can arrive at the result in the following two ways.

Proof 1: Involution Method:-

The problem here is one of involution, that is raising a quantity to a power. To raise a quantity to a second power is to multiply the quantity by itself. In the given problem, the quantity is (x + y). Therefore,

(x + y)² = (x+y) (x+y)

Multiplying term by term

(x + y)² = x² + xy + yx + y²

Using the commutative property of algebra,

xy = yx

Hence, (x + y)² = x² + xy + xy + y²

Or, (x + y)² = x² + 2xy + y² which proves our assertion

Proof 2: Binomial Method:-

The Binomial Theorem states that

(x + a)^n = x^n + n a x^(n-1) + [n(n-1)/1.2] a² x^(n-2) + ………

Now put a = y and n = 2 (number of terms = n+1 = 3) to obtain

(x + y)² = x² + 2 y x^(2-1) + (2.1/1.2) y² x^(0) + 0

which on simplification gives,

(x + y)² = x² + 2 y.x¹ + 1. y². 1 (since x^(0) = 1)

Or, (x + y)² = x² + 2xy + y² (Proved)

Answer 2

Answer:- (x + y)² = x² + 2xy + y²

The square of the sum of two quantities is equal to the square of the first term, plus twice the product of the two terms, plus the square of the second term.

We can arrive at the result in the following two ways.

Proof 1: Involution Method:-

The problem here is one of involution, that is raising a quantity to a power. To raise a quantity to a second power is to multiply the quantity by itself. In the given problem, the quantity is (x + y). Therefore,

(x + y)² = (x+y) (x+y)

Multiplying term by term

(x + y)² = x² + xy + yx + y²

Using the commutative property of algebra,

xy = yx

Hence, (x + y)² = x² + xy + xy + y²

Or, (x + y)² = x² + 2xy + y² which proves our assertion

Proof 2: Binomial Method:-

The Binomial Theorem states that

(x + a)^n = x^n + n a x^(n-1) + [n(n-1)/1.2] a² x^(n-2) + ………

Now put a = y and n = 2 (number of terms = n+1 = 3) to obtain

(x + y)² = x² + 2 y x^(2-1) + (2.1/1.2) y² x^(0) + 0

which on simplification gives,

(x + y)² = x² + 2 y.x¹ + 1. y². 1 (since x^(0) = 1)

Or, (x + y)² = x² + 2xy + y² (Proved)