Q. 3 E5.0( 1 Vote )

# Factorize the following expressions:

25x^{2} + 4y^{2} + 9z^{2} – 20xy + 12yz – 30zx

Answer :

__Method 1__

The above equation can be simplified as :

= (– 5x)^{2} + (2y)^{2} + (3z)^{2} + 2(– 5x)(2y) + 2(2y)(3z) + 2(3z)(– 5x)

This equation is of the form: p^{2} + q^{2} + r^{2} + 2pq + 2qr + 2rp

where p = – 5x, q = 2y, r = 3z

Using the identity: p^{2} + q^{2} + r^{2} + 2pq + 2qr + 2rp = (p + q + r)^{2}

We get,

⇒ 25x^{2} + 4y^{2} + 9z^{2} – 20xy + 12yz – 30zx = (– 5x + 2y + 3z)^{2}

= (– 5x + 2y + 3z)^{2}

= (– 1)^{2}(5x – 2y – 3z)^{2}

= (5x – 2y – 3z)^{2}

__Method 2__

The above equation can be simplified as:

= (– 5x)^{2} + (2y)^{2} + (3z)^{2} + 2(– 5x)(2y) + 2(2y)(3z) + 2(3z)(– 5x)

= {(– 5x)^{2} + 2(– 5x)(2y) + (2y)^{2}} + (3z)^{2} + 2(2y)(3z) + 2(3z)(– 5x)

(∵ p^{2} + 2pq + q^{2} = (p + q)^{2})

= (– 5x + 2y)^{2} + (3z)^{2} + 2(2y)(3z) + 2(3z)(– 5x)

Taking 2(3z) common in term 2(2y)(3z) + 2(3z)(– 5x)

= (– 5x + 2y)^{2} + (3z)^{2} + 2(3z)(2y – 5x)

= (– 5x + 2y)^{2} + 2(3z)(2y – 5x) + (3z)^{2}

This of the form: (p + q)^{2} + 2r(p + q) + r^{2}

Where, p = – 5x, q = 2y and r = 3z

Using the identity: (p + q)^{2} = p^{2} + 2pq + q^{2}

We get,

⇒ 25x^{2} + 4y^{2} + 9z^{2} – 20xy + 12yz – 30zx = (– 5x + 2y + 3z)^{2}

= (– 5x + 2y + 3z)^{2}

= (– 1)^{2}(5x – 2y – 3z)^{2}

= (5x – 2y – 3z)^{2}

25x^{2} + 4y^{2} + 9z^{2} – 20xy + 12yz – 30zx = (5x – 2y – 3z)^{2}

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