How do you add #\frac { 3t } { t + 5} + \frac { 32t + 10} { t ^ { 2} - 25}#?

1 Answer
Jim G.
Mar 20, 2017

#(3t+2)/(t-5)#

Explanation:

Before adding/subtracting fractions they must have a #color(blue)"common denominator."# At the moment they don't.

#t^2-25" is a " color(blue)"difference of squares"#

#"and factorises as " (t+5)(t-5)#

Fractions may be expressed as.

#(3t)/(t+5)+(32t+10)/((t+5)(t-5))#

multipling numerator/denominator of #(3t)/(t+5)" by"#

#color(blue)((t-5)/(t-5))# creates a common denominator for the fractions.

#rArr((3t)(t-5))/((t+5)(t-5))+(32t+10)/((t+5)(t-5))#

Now the fractions have a common denominator we can add the numerators leaving the denominator as it is.

#rArr(3t^2-15t+32t+10)/((t+5)(t-5))=(3t^2+17t+10)/((t+5)(t-5))#

We can reduce the fraction further by factorising #3t^2+17t+10# and then #color(blue)"cancelling common factors"#

#3t^2+17t+10=(3t+2)(t+5)#

#rArr((3t+2)cancel((t+5)^1))/(cancel((t+5)^1)(t-5)#

#=(3t+2)/(t-5)#

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