Question 1209214
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Consider the following calculation, using the pattern {{{x+y)^2=x^2+2xy+y^2}}}:<br>
{{{(sqrt(a)+sqrt(b))^2=a+2sqrt(ab)+b=(a+b)+2sqrt(ab)}}}<br>
To evaluate the given expression, put the expression in the radical in the form above by extracting {{{sqrt(4)=2}}} from the radical:<br>
{{{sqrt((80)+sqrt(1776))=sqrt((80)+2sqrt(444))}}}<br>
The problem is now finished by finding a and b such that (a+b)=80 and ab=444.<br>
Solve that pair of equations to find the answer, remembering that the problem specifies a < b.<br>
{{{a+b = 80}}}
{{{b = 80-a}}}
{{{ab = a(80-a)=444}}}
{{{80a-a^2=444=0}}}
{{{a^2-80a+444=0}}}<br>
Note that to solve this by factoring (without using the quadratic formula), we need to find two numbers whose product is 444 and whose sum is 80.  That is what the original problem requires us to do, so the formal algebra doesn't get us any closer to the answer.<br>
So use logical trial and error to find the two numbers are 74 and 6: 74+6 = 80 and 74*6 = 444.<br>
Then, since a is the smaller of the two numbers....<br>
ANSWER: a = 6, b = 74; {{{sqrt((80)+sqrt(1776))=sqrt(6)+sqrt(74)}}}<br>