Question 174257
Call the 1st integer {{{n}}}
The next consecutive integer will be {{{n+1}}}
{{{n^2 + (n + 1)^2 = 41}}}
{{{n^2 + n^2 + 2n + 1 = 41}}}
{{{2n^2 + 2n - 40 = 0}}}
{{{n^2 + n - 20 = 0}}}
Use complete the square method
{{{n^2 + n = 20}}}
{{{n^2 + n + (1/2)^2 = 20 + (1/2)^2}}}
{{{(n + (1/2))^2 = (80 + 1)/4}}}
Take the square root of both sides
{{{n + 1/2 = 9/2}}}
{{{n = 8/2}}}
{{{n = 4}}}
{{{n + 1 = 5}}}
The numbers are 4 and 5
check answer:
{{{n^2 + (n + 1)^2 = 41}}}
{{{4^2 + 5^2 = 41}}}
{{{16 + 25 = 41}}}
{{{41 = 41}}}
OK
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Let {{{w}}} = width
Let {{{l}}} =length
Let {{{A}}} =area
given:
{{{l = w + 3}}}
{{{A = l*w}}}
{{{A = 152}}}in2
{{{152 = l*w}}}
{{{152 = (w+3)*w}}}
{{{w^2 + 3w - 152 = 0}}}
Use complete the square
{{{w^2 + 3w = 152}}}
{{{w^2 + 3w + (3/2)^2 = 152 + (3/2)^2}}}
{{{(w + 3/2)^2 = (608 + 9)/4}}}
{{{(w + 3/2)^2 = 617/4}}}
{{{w + 3/2 = sqrt(617)/2}}}
{{{w = (sqrt(617) - 3)/2}}}