Question 59202
Let the consecutive integers equal x and x+1
{{{x^2 + (x+1)^2 = 85}}}
{{{2x^2 + 2x + 1 = 85}}}
This can be solved either by the quadratic formula
of by completing the square. I'll complete the square
subtract 1 from both sides
{{{2x^2 + 2x = 84}}}
divide both sides by 2
{{{x^2 + x = 42}}}
to complete the square, take one-half of the coeffecient
of x, square it, and add it to both sides
Half of 1 is 1/2 and (1/2)^2 = 1/4
{{{x^2 + x + 1/4 = 42 + 1/4}}}
{{{(x + 1/2)^2 = (168 + 1)/ 4}}}
{{{x + 1/2 = (0 +- sqrt(169)) / 2}}}
{{{x = +13/2 - 1/2}}}
{{{x = +12/2}}}
{{{x = +6}}}
and
{{{x = -13/2 - 1/2}}}
{{{x = -14/2}}}
{{{x = -7}}}
The problem wants 2 consecutive POSITIVE integers, so I'll
pick x = +6
{{{x + 1 = +7}}}
The answers are +6 and +7
check:
{{{6^2 + 7^2 = 85}}}
{{{36 + 49 = 85}}}
{{{85 = 85}}}
OK