How do you write the following expression in standard form #(2i)/(2+i)+5/(2-i)#?

1 Answer
Oct 23, 2016

Answer:

Combine the two fractions using a common denominator and then simplify. #12/5 + 9/5i#

Explanation:

Given:
#(2i)/(2 + i) + 5/(2 - i) =#

Make the common denominator #(2 + i)(2 - i)#

#(2i)/(2 + i)((2 - i)/(2 - i)) + 5/(2 - i)((2 + i)/(2 + i)) =#

Combine over the denominator:

#((2i)(2 - i) + 5(2 + i))/((2 - i)(2 + i)) =#

Multiply the denominator, using the pattern #(a + b)(a - b) = a^2 - b^2#:

#((2i)(2 - i) + 5(2 + i))/(4 - i^2) =#

Substitute + 1 for #-i^2#

#((2i)(2 - i) + 5(2 + i))/(4 + 1) =#

#((2i)(2 - i) + 5(2 + i))/5 =#

Perform the multiplications in the numerator:

#(4i - 2i^2 + 10 + 5i)/5 =#

Replace #-2i^2# we +2:

#(4i + 2 + 10 + 5i)/5 =#

Combine like terms:

#(12 + 9i)/5 =#

#12/5 + 9/5i#

Standard form #a + bi#