Question

Find square root of x^4+2x^3-x+(1/4)

Answer 1

ANSWER:

(x² + x - 1/2)

GIVEN:

x⁴ + 2x³ - x+ (1/4)

TO FIND:

The square root of x⁴ + 2x³ - x+ (1/4)

EXPLANATION:

$\: \: \: \quad \: \quad &\qquad \qquad \: \: \: x^2+x-\frac{1}{2}\\ \begin{array}{c|c} \cline{2 - 2} x^2&x^4+2x^3-x+\frac{1}{4}\\&x^4\qquad \qquad \qquad\\&(-)\qquad \qquad \qquad\\ \cline{2-2}2x^2+x&2x^3-0x^2\\&2x^3+x^2\\&(-)\: \:(-)\\ \cline{2-2} 2x^2+2x-\frac{1}{2}&-x^2-x+\frac{1}{4}\\ &-x^2-x+\frac{1}{4}\\ &(+) \: (+) \: (-) \\ \cline{2-2} &0 \end{array}$

HENCE

$\bf{ \sqrt{x^4+2x^3-x+ \frac{1}{4} } = {x}^{2} + x - \frac{1}{2} }$

VERIFICATION:

(x² + x - 1/2)(x² + x - 1/2)

x⁴ + x³ - 1/2 x² + x³ + x² - 1/2 x - 1/2 x² - 1/2x + 1/4

x⁴ + x³ + x³ - 1/2 x²- 1/2 x² + x² - 1/2 x - 1/2x + 1/4

x⁴ + 2x³ - x² + x² - x + 1/4

x⁴ + 2x³ - x + 1/4 = (x² + x - 1/2)(x² + x - 1/2)

HENCE VERIFIED.