Question

Find the square root of $x^4-10x^3+37x^2-60x+36$.

• A. $|(x^2-5x+6)|$
• B. $|(x^2+5x+6)|$
• C. $|(x^2-5x-4)|$
• D. None of these

Answer 1

The correct option is A $|(x^2-5x+6)|$

To find the square root of the given polynomial $x^4-10x^3+37x^2-60x+36$, we must equate it to the general form of equation that is $(a{x}^2+bx+c)$ as shown below:

$\sqrt{x^4-10x^3+37x^2-60x+36}=a{x}^2+bx+c\Rightarrow {\left(\sqrt{x^4-10x^3+37x^2-60x+36}\right)}^2={(a{x}^2+bx+c)}^2\Rightarrow x^4-10x^3+37x^2-60x+36={(a{x}^2+bx+c)}^2\Rightarrow x^4-10x^3+37x^2-60x+36={\left(a{x}^2\right)}^2+{\left(bx\right)}^2+{\left(c\right)}^2+(2\times a{x}^2\times bx)+(2\times bx\times c)+(2\times c\times a{x}^2)(\because (a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca)\Rightarrow x^4-10x^3+37x^2-60x+36=a^2{x}^4+b^2{x}^2+c^2+2ab{x}^3+2bcx+2ac{x}^2$

Now, comparing the coefficients, we get:

$a^2=1,b^2+2ac=37,c^2=36,2ab=-10,2bc=-60$

$a^2=1\Rightarrow a=1$

$c^2=36\Rightarrow c=6$

$2ab=-10\Rightarrow 2\times 1\times b=-10\Rightarrow 2b=-10\Rightarrow b=-5$

Therefore, $a=1,b=-5,c=6$ and substituting the values in the equation $(a{x}^2+bx+c)$, we get $x^2-5x+6$.

Hence, the square root of $x^4-10x^3+37x^2-60x+36$ is $\left|\right(x^2-5x+6\left)\right|$.