Question 149914
{{{(a)/(ac-c^2)-(c)/(a^2-ac)}}} Start with the given expression.



{{{(a)/(c(a-c))-(c)/(a^2-ac)}}} Factor {{{ac-c^2}}} to get {{{c(a-c)}}}



{{{(a)/(c(a-c))-(c)/(a(a-c))}}} Factor {{{a^2-ac}}} to get {{{a(a-c)}}}



Notice how the LCD is {{{ac(a-c)}}}



{{{(a/a)((a)/(c(a-c)))-(c)/(a(a-c))}}} Multiply the first fraction by {{{a/a}}}. This will make the first fraction have a denominator of {{{ac(a-c)}}}



{{{(a^2)/(ac(a-c))-(c)/(a(a-c))}}} Combine the fractions and multiply.



{{{(a^2)/(ac(a-c))-(c/c)((c)/(a(a-c)))}}} Multiply the second fraction by {{{c/c}}}. This will make the second fraction have a denominator of {{{ac(a-c)}}}


 
{{{(a^2)/(ac(a-c))-(c^2)/(ac(a-c))}}} Combine the fractions and multiply.



Since the denominators are now equal, this means that we can combine the fractions.


{{{(a^2-c^2)/(ac(a-c))}}} Combine the numerators over the common denominator.



{{{((a+c)(a-c))/(ac(a-c))}}}  Factor {{{a^2-c^2}}} to get {{{(a+c)(a-c)}}}



{{{((a+c)highlight((a-c)))/(ac*highlight((a-c)))}}} Highlight the common terms.



{{{((a+c)cross((a-c)))/(ac*cross((a-c)))}}} Cancel out the common terms.



{{{(a+c)/(ac)}}} Simplify



So {{{(a)/(ac-c^2)-(c)/(a^2-ac)}}} simplifies to {{{(a+c)/(ac)}}}.