How do you divide #\frac { 5a ^ { 2} + 49a - 10} { 14a ^ { 2} + 30a + 4} \div \frac { 5a ^ { 2} - 51a + 10} { 14a ^ { 2} + 30a + 4}#?

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Answer 1 · 2018-01-16T18:20:48

#(a+10)/(a-10)#

To begin flip the second fraction upside-down to turn the divide into a multiply:

#(5a^2+49a-10)/(14a^2+30a+4)divide(5a^2-51a+10)/(14a^2+30a+4)#

#=(5a^2+49a-10)/(14a^2+30a+4)times(14a^2+30a+4)/(5a^2-51a+10)#

Straight away we can cancel the denominator of the first with the numerator of the second like so:

#=(5a^2+49a-10)/cancel(14a^2+30a+4)timescancel(14a^2+30a+4)/(5a^2-51a+10)#

#=(5a^2+49a-10)/1times1/(5a^2-51a+10)#

#=(5a^2+49a-10)/(5a^2-51a+10)#

These polynomials can be factorised:

#5a^2+49a-10=(5a-1)(a+10)#

and

#5a^2-51a+10=(5a-1)(a-10)#

So the fraction now becomes:

#=(5a^2+49a-10)/(5a^2-51a+10)=((5a-1)(a+10))/((5a-1)(a-10))#

We can now cancel the #5a-1# to leave us with:

#(cancel((5a-1))(a+10))/(cancel((5a-1))(a-10))=(a+10)/(a-10)#