Example

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

Question

$$\frac{ 4a-1 }{ { a }^{ 2 } -16 } \times \frac{ { a }^{ 2 } -4a }{ 2a+1 }$$

Answer

(a*(4*a-1))/((a+4)*(2*a+1))

Solution


Rewrite \({a}^{2}-16\) in the form \({a}^{2}-{b}^{2}\), where \(a=a\) and \(b=4\).
\[\frac{4a-1}{{a}^{2}-{4}^{2}}\times \frac{{a}^{2}-4a}{2a+1}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{4a-1}{(a+4)(a-4)}\times \frac{{a}^{2}-4a}{2a+1}\]
Factor out the common term \(a\).
\[\frac{4a-1}{(a+4)(a-4)}\times \frac{a(a-4)}{2a+1}\]
Cancel \(a-4\).
\[\frac{4a-1}{a+4}\times \frac{a}{2a+1}\]
Use this rule: \(\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\).
\[\frac{(4a-1)a}{(a+4)(2a+1)}\]
Regroup terms.
\[\frac{a(4a-1)}{(a+4)(2a+1)}\]
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