Question 87338
Let x=1st #, y=2nd #, z=3rd #


{{{x^2+y^2+z^2=77}}} This is the translation of "the sum of their (each number) squares is 77"


Since each number is consecutive, we're really adding 1 each time. So 


{{{y=x+1}}} and {{{z=(x+1)+1=x+2}}}



{{{x^2+(x+1)^2+(x+2)^2=77}}} Plug in {{{y=x+1}}} and {{{z=x+2}}}


{{{x^2+(x^2+2x+1)+(x^2+4x+4)=77}}} Foil


{{{3x^2+6x+5=77}}} Combine like terms


{{{3x^2+6x+5-77=0}}} Subtract 77 from both sides


{{{3x^2+6x-72=0}}} Combine like terms


Now lets use the quadratic formula to solve for x



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general form of the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}


So lets solve {{{3*x^2+6*x-72=0}}}


{{{x = (-6 +- sqrt( (6)^2-4*3*-72 ))/(2*3)}}} Plug in a=3, b=6, and c=-72




{{{x = (-6 +- sqrt( 36-4*3*-72 ))/(2*3)}}} Square 6 to get 36




{{{x = (-6 +- sqrt( 36+864 ))/(2*3)}}} Multiply {{{-4*-72*3}}} to get {{{864}}}




{{{x = (-6 +- sqrt( 900 ))/(2*3)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-6 +- 30)/(2*3)}}} Simplify the square root




{{{x = (-6 +- 30)/6}}} Multiply 2 and 3 to get 6


So now the expression breaks down into two parts


{{{x = (-6 + 30)/6}}} or {{{x = (-6 - 30)/6}}}


Lets look at the first part:


{{{x=24/6}}} Add the terms in the numerator

{{{x=4}}} Divide


So one answer is

{{{x=4}}}

Now lets look at the second part:


{{{x=-36/6}}} Subtract the terms in the numerator

{{{x=-6}}} Divide


So another answer is

{{{x=-6}}}


So our solutions are:

{{{x=4}}} or {{{x=-6}}}


Notice when we graph {{{3*x^2+6*x-72}}} we get:


{{{ graph( 500, 500, -16, 14, -16, 14,3*x^2+6*x+-72) }}}


and we can see that the roots are {{{x=4}}} and {{{x=-6}}}. This verifies our answer


So our first number could be 4 or -6


Lets use {{{x=4}}} to find the 2nd and 3rd number

{{{y=4+1}}} and {{{z=4+2}}}
{{{y=5}}} and {{{z=6}}}

So we have 3 numbers: 4,5,6


Check:
{{{4^2+5^2+6^2=77}}}
{{{16+25+36=77}}}
{{{77=77}}} works


So the 3 numbers 4,5,6 work



Now lets use {{{x=-6}}} to find the 2nd and 3rd number

{{{y=-6+1}}} and {{{z=-6+2}}}
{{{y=-5}}} and {{{z=-4}}}

So we have 3 numbers: -6,-5,-4


Check:
{{{(-6)^2+(-5)^2+(-4)^2=77}}}
{{{36+25+16=77}}}
{{{77=77}}} works


So the 3 numbers -6,-5,-4 work


Answer:

So if you allow negative numbers, you get 2 possible answers

4,5,6

or 

-6,-5,-4