Question 236845
Remember, perfect squares are of the form {{{(a+b)^2}}}


In this case, we want a perfect square of the form {{{(r+x)^2}}} where 'x' is an unknown expression or number.



{{{(r+x)^2=r^2+10r+k}}} Set that perfect square expression equal to the given expression.



{{{r^2+2rx+x^2=r^2+10r+k}}} FOIL the left side.



{{{2rx+x^2=10r+k}}} Subtract {{{r^2}}} from both sides.



Since the only terms with 'r' in them are 2rx and 10r, this means that {{{2rx=10r}}}. Divide both sides by 'r' to get {{{2x=10}}}. Now solve for 'x' to get {{{x=5}}}


So the left side of {{{(r+x)^2}}} becomes {{{(r+5)^2}}}. FOIL this expression out to get {{{(r+5)^2=r^2+10r+25}}}. So we see that the value of k is k=25.



Hopefully you understand this whole process (if not, just ask). Once you do, there is a handy shortcut. The value of 'k' is simply equal to half the coefficient of 'r' squared. In other words,


{{{k=(10/2)^2=5^2=25}}} which is what we just got.