With a rational denominator, the expression 2 2 + 3 - 5 is equivalent to:

Let k=2+3 Then 2k-5 2(k+5)k^2-5^22k+10k^2-5 2(2+3)+10(2+3)^2-5 2+6+102626+6+6012 6+3+156 3 + 6 + 156

smc

algebra senior

Problem

With a rational denominator, the expression

\frac{2}{2 + 3 - 5}

is equivalent to:

(A)

\frac{3 + 6 + 15}{6}

(B)

\frac{6 - 2 + 10}{6}

(C)

\frac{2 + 6 + 10}{10}

(D)

\frac{2 + 6 - 10}{6}

Solution

Let

k = 2 + 3

Then

\frac{2}{k - 5} ⟹ \frac{2 ( k + 5 )}{k ^{2} - 5 ^{2}} ⟹ \frac{2 k + 10}{k ^{2} - 5} ⟹ \frac{2 ( 2 + 3 ) + 10}{( 2 + 3 ) ^{2} - 5} ⟹ \frac{2 + 6 + 10}{2 6} ⟹ \frac{2 6 + 6 + 60}{12} ⟹ \frac{6 + 3 + 15}{6} \frac{3 + 6 + 15}{6}

Final answer