Question

21. What should be added to make each of the following a perfect square-81x2-18x
​

Answer 1

$\textbf{Given:}$

$-81x^2-18x$

$\textbf{To find:}$

$\text{The term should be added to make the given polynomial as a perfect square}$

$\textbf{Solution:}$

$-81x^2-18x$

$\text{Take out -81 as common factor}$

$=-81[x^2+\frac{18}{81}x]$

$=-81[x^2+\frac{2}{9}x]$

$\text{Squaring and adding half of the coefficient of x}$

$=-81[x^2+\frac{2}{9}x+(\frac{1}{9})^2]$

$=-81[x^2+2(x)(\frac{1}{9})+(\frac{1}{9})^2]$

$\text{Using the identity}$

$\boxed{\bf\,a^2+2\,ab+b^2=(a+b)^2}$

$=-81(x+\frac{1}{9})^2$

$\text{Now,}\;-81(x+\frac{1}{9})^2$

$=-81[x^2+2(1)(\frac{1}{9}x)+(\frac{1}{9})^2]$

$=-81x^2-18x-1$

$\therefore\textbf{-1 should be addded to make the given polynomial as a perfect square}$

Find more:

What should be subtracted from the polynomial x + 2x + 8x +22x + 18 so that it is completely divisible

by x2 + 6 ?​

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