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Class 9
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Math
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Algebraic Manipulation
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Square root of an algebraic expression

Algebraic Manipulation

Square root of an algebraic expression

Math

Square root of algebraic expression by Factorization:

To find the square root of an algebraic expression by factorization, we need to factor the expression into its irreducible factors and then take the square root of each factor.

Example: Find the square root of 36(3−2x)2−48(3−2x)y+16y236(3-2x)^2 - 48(3-2x)y+16y^236(3−2x)2−48(3−2x)y+16y2.

Solution:

36(3−2x)2−48(3−2x)y+16y236(3-2x)^2 - 48(3-2x)y+16y^236(3−2x)2−48(3−2x)y+16y2

                                      =(6(3−2x))2−2(6(3−2x))(4y)+(4y)2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= (6(3-2x))^2 -2(6(3-2x))(4y) + (4y)^2=(6(3−2x))2−2(6(3−2x))(4y)+(4y)2

Using, a2−2ab+b2=(a−b)2a^2-2ab+b^2 = (a-b)^2a2−2ab+b2=(a−b)2, we get,

                                      =(6(3−2x)−4y)2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= (6(3-2x) - 4y)^2=(6(3−2x)−4y)2

                                      =(18−12x−4y)2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= (18-12x - 4y)^2=(18−12x−4y)2

Now taking square root on both sides, we get,

                                      =(18−12x−4y)2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= (18-12x - 4y)^2=(18−12x−4y)2

                                      =(18−12x−4y)2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \sqrt{(18-12x - 4y)^2}=(18−12x−4y)2​

                                      =18−12x−4y\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= 18-12x - 4y=18−12x−4y

Hence, the square root of 36(3−2x)2−48(3−2x)y+16y236(3-2x)^2 - 48(3-2x)y+16y^236(3−2x)2−48(3−2x)y+16y2 is 18−12x−4y18-12x - 4y18−12x−4y.

Square root of algebraic expression by Division Method: The procedure for calculating square root by division method is almost the same as of the numbers. Let's discuss it with the help of an example.

Example:

Find the square root of 9x4+12x3+4x29x^4 + 12x^3 + 4x^29x4+12x3+4x2 using the Division Method.

Solution:

Try to find a square of a term that equals 9x49x^49x4, which is 3x23x^23x2,

Now try to add something 3x23x^23x2 and multiply it with the same number to get 12x312 x^312x3,

Hence the square root of 9x4+12x3+4x29x^4 + 12x^3 + 4x^29x4+12x3+4x2 is 3x2+2x3x^2+2x3x2+2x.

Example: Find the square root of x2−2x+3−2x+1x2x^2-2x+3 - \frac{2}{x} + \frac{1}{x^2}x2−2x+3−x2​+x21​.

Solution:

Thus the square root of x2−2x+3−2x+1x2x^2-2x+3 - \frac{2}{x} + \frac{1}{x^2}x2−2x+3−x2​+x21​ is x−1+1xx-1 + \frac{1}{x}x−1+x1​.