Square root of an algebraic expression

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 $36(3-2x)^2 - 48(3-2x)y+16y^2$.

Solution:

$36(3-2x)^2 - 48(3-2x)y+16y^2$

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

Using, $a^2-2ab+b^2 = (a-b)^2$, we get,

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

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

Now taking square root on both sides, we get,

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

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

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

Hence, the square root of $36(3-2x)^2 - 48(3-2x)y+16y^2$ is $18-12x - 4y$.

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 $9x^4 + 12x^3 + 4x^2$ using the Division Method.

Solution:

Try to find a square of a term that equals $9x^4$, which is $3x^2$,

Now try to add something $3x^2$ and multiply it with the same number to get $12 x^3$,

Hence the square root of $9x^4 + 12x^3 + 4x^2$ is $3x^2+2x$.

Example: Find the square root of $x^2-2x+3 - \frac{2}{x} + \frac{1}{x^2}$.

Solution:

Thus the square root of $x^2-2x+3 - \frac{2}{x} + \frac{1}{x^2}$ is $x-1 + \frac{1}{x}$.