
Example: Find the square root of x^{4}  4x^{3} + 8x^{2}  8x + 4. Solution: The given expression being of degree four, its square root will be a quadratic expression. Therefore we may assume x^{4}  4x^{3} + 8x^{2}  8x + 4 = (x^{} + lx + m)^{2}, where l and m are to be determined. (x^{} + lx + m)^{2} = (x^{} + lx + m) (x^{} + lx + m) = x^{4} + lx^{3} + mx^{2} + lx^{3} + l^{2}x^{2} + lmx + mx^{2} + lmx + m^{2} = x^{4} + 2lx^{3} + 2mx^{2} + l^{2}x^{2} + 2lmx + m^{2} = x^{4} + 2lx^{3} + (l^{2} + 2m)x^{2} + 2lmx + m^{2} Thus, x^{4}  4x^{3} + 8x^{2}  8x + 4 = x^{4} + 2lx^{3} + (l^{2} + 2m)x^{2} + 2lmx + m^{2} Comparing the coefficients of powers of x on either sides, we get 2l = 4, l^{2} + 2m = 8, 2lm = 8, m^{2} = 4 Therefore, l = 2 and m = 2. Therefore, the square root is x^{2} + lx + m = x^{2}  2x + 2. Directions: Solve the following problems using above method. Also write at least 5 examples of your own. 