Example

3(r+2s)=2t-4
a⋅(b-2)=3b
3(r+2s)=2t-4
a⋅(b-2)=3b

Question

$$\frac{3x^{2}-3x}{x^{2}-6x+5}:\frac{x^{2}-25}{6x}; 8+x^{3}+\frac{3x}{x-25}=$$

Answer

$$(3*x*(x-1))/((x-5)*(x-1))/(((x+5)*(x-5))/(6*x));8+x^3+(3*x)/(x-25)$$

Solution


Factor out the common term \(3x\).
\[\begin{aligned}&\frac{\frac{3x(x-1)}{{x}^{2}-6x+5}}{\frac{{x}^{2}-25}{6x}}\\&8+{x}^{3}+\frac{3x}{x-25}\end{aligned}\]
Factor \({x}^{2}-6x+5\).
\[\begin{aligned}&\frac{\frac{3x(x-1)}{(x-5)(x-1)}}{\frac{{x}^{2}-25}{6x}}\\&8+{x}^{3}+\frac{3x}{x-25}\end{aligned}\]
Rewrite \({x}^{2}-25\) in the form \({a}^{2}-{b}^{2}\), where \(a=x\) and \(b=5\).
\[\begin{aligned}&\frac{\frac{3x(x-1)}{(x-5)(x-1)}}{\frac{{x}^{2}-{5}^{2}}{6x}}\\&8+{x}^{3}+\frac{3x}{x-25}\end{aligned}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\begin{aligned}&\frac{\frac{3x(x-1)}{(x-5)(x-1)}}{\frac{(x+5)(x-5)}{6x}}\\&8+{x}^{3}+\frac{3x}{x-25}\end{aligned}\]
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