Question 1: Factorize the given expression: 9x^{2} + 49y^{2} + 25z^{2} - 42xy - 30xz + 70yz

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The coefficient of xy and xz, -42 and -30, are negative and the coefficient of yz, 70, is positive.

So, we can deduce that the coefficient of x is negative and the coefficients of y and z are positive.

Alternatively, the coefficient of x is positive and those of y and z are negative.

Both answers are correct.

The given expression mimics the algebraic identity (a + b + c)^{2}

(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca

9x^{2} + 49y^{2} + 25z^{2} - 42xy - 30xz + 70yz

= (-3x)^{2} + (7y)^{2} + (5z)^{2} + 2(-3x)(7y) + 2(-3x)(5z) + 2(7y)(5z)

(-3x)^{2} + (7y)^{2} + (5z)^{2} + 2(-3x)(7y) + 2(-3x)(5z) + 2(7y)(5z) resembles a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca

∴ (-3x)^{2} + (7y)^{2} + (5z)^{2} + 2(-3x)(7y) + 2(-3x)(5z) + 2(7y)(5z) = (-3x + 7y + 5z)^{2}

= (-3x + 7y + 5z)(-3x + 7y + 5z)

Class 9 Maths

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