Question 187552
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First Condition: "Sum of three consecutive positive integers"
{{{x^2+(x+1)^2+(x+2)^2}}}
Second Condition: "Equal to the sum of the next two integers"
{{{(x+3)^2+(x+4)^2}}}


Equating the two conditions being equal:
{{{x^2+(x+1)^2+(x+2)^2=(x+3)^2+(x+4)^2}}}, Equation 1
Expand:
{{{x^2+x^2+2x+1+x^2+4x+4=x^2+6x+9+x^2+8x+16}}}
Addition/Subtraction (Cancellations):
{{{cross(x^2)+cross(x^2)+6x+x^2+5=cross(x^2)+14x+25+cross(x^2)}}}
Combine  terms:
{{{x^2+6x-14x+5-25=0}}}---->{{{x^2-8x-20=0}}}
Factorable being perfect square:---><font color=red>(x-10)</font>(x+2)=0
Use highlighted, x=<font color=red>10</font>, 1st Positive Integer, and follows the next four= <font color=red>11,12,13,14</font>

Go back Eqn 1 to verify:
{{{10^2+11^2+12^2=13^2+14^2}}}
{{{100+121+144=169+196}}}
{{{365=365}}}
Thank you,
Jojo</font>