Question

Ctiny%7B%21%7D%7D%7D%20%7D%7B%201-3x%20%7D" id="TexFormula1" title="5x- \dfrac{ \sqrt{ 9 { x }^{ 2 } -6x+1 \phantom{\tiny{!}}} }{ 1-3x }" alt="5x- \dfrac{ \sqrt{ 9 { x }^{ 2 } -6x+1 \phantom{\tiny{!}}} }{ 1-3x }" align="absmiddle" class="latex-formula">

Answer 1

Answer:

$=5x+1$          or          $=5x-1$

Step-by-step explanation:

One is given the following equation:

$5x-\frac{\sqrt{9x^2-6x+1}}{1-3x}$

The problem asks one to simplify the expression, the first step in solving this equation is to factor the equation.  Rewrite the numerator and denominator of the fraction as the product of two expressions. Remember the factoring patterns:

$(a-b)^2=a^2-2ab+b^2$

$=5x-\frac{\sqrt{9x^2-6x+1}}{1-3x}$

$=5x-\frac{\sqrt{(3x-1)^2}}{-(3x-1)}$

Now simplify the numerator. Remember, taking the square root of a squared value is the same as taking the absolute value of the expression,

$=5x-\frac{\sqrt{(3x-1)^2}}{1-3x}$

$=5x-\frac{|3x-1|}{-(3x-1)}$

Rewrite the expression without the absolute value sign in the numerator. Remember the general rule for removing the absolute value sign:

$|a-b|\\=a -b$           or           $(-a-b) = b-a$

$=5x-\frac{|3x-1|}{-(3x-1)}$

$=5x-\frac{3x-1}{-(3x-1)}$          or          $=5x-\frac{-(3x-1)}{-(3x-1)}$

Simplify both expressions, reduce by canceling out common terms in both the numerator and the denominator,

$=5x-\frac{3x-1}{-(3x-1)}$          or          $=5x-\frac{-(3x-1)}{-(3x-1)}$

$=5x-(-1)$          or          $=5x-(1)$

Simplify further by rewriting the expression without the parenthesis, remember to distribute the sign outside the parenthesis by the terms inside of the parenthesis; note that negative times negative equals positive.

$=5x-(-1)$          or          $=5x-(1)$

$=5x+1$          or          $=5x-1$