Factorize the following expressions:
9x2 + y2 + 1 – 6xy + 6x – 2y
Method 1
The above equation can be simplified as :
= (3x)2 + (– y)2 + (1)2 + 2(3x)(– y) + 2(3x)(1) + 2(– y)(1)
This equation is of the form: p2 + q2 + r2 + 2pq + 2qr + 2rp
where p =3x, q = – y, r = 1
Using the identity: p2 + q2 + r2 + 2pq + 2qr + 2rp = (p + q + r)2
We get,
⇒ 9x2 + y2 + 1 – 6xy + 6x – 2y = (3x – y + 1)2
= (3x – y + 1)2
Method 2
The above equation can be simplified as:
= (3x)2 + (– y)2 + (1)2 + 2(3x)(– y) + 2(3x)(1) + 2(– y)(1)
= {(3x)2 + 2(3x)(– y) + (– y)2} + (1)2 + 2(3x)(1) + 2(– y)(1)
(∵ p2 + 2pq + q2 = (p + q)2)
= (3x – y)2 + (1)2 + 2(3x)(1) + 2(– y)(1)
Taking 2(1) common in term 2(3x)(1) + 2(– y)(1)
= (3x – y)2 + (1)2 + 2(1)(3x – y)
= (3x – y)2 + 2(1)(3x – y) + (1)2
This of the form: (p + q)2 + 2r(p + q) + r2
Where, p = 3x, q = – y and r = 1
Using the identity: (a + b)2 = a2 + 2ab + b2
We get,
⇒ 9x2 + y2 + 1 – 6xy + 6x – 2y = (3x – y + 1)2
9x2 + y2 + 1 – 6xy + 6x – 2y = (3x – y + 1)2
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