How do you find the value of c that makes #x^2+5x+c# into a perfect square?

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Answer 1 · 2017-02-20T19:48:57

#c = 25/4#

#color(white)()#
Method 1

Note that #(x+k)^2 = x^2+2k+k^2#

So if #2k = 5# then #k = 5/2# and:

#(x+5/2)^2 = x^2+5x + (5/2)^2 = x^2+5x+25/4#

So #c = 25/4#

#color(white)()#
Method 2

#x^2+5x+c#

is in the form:

#ax^2+bx+c#

with #a=1# and #b=5#.

This has discriminant #Delta# given by the formula:

#Delta = b^2-4ac = 25-4c#

So if #Delta = 0# (indicating a repeated zero) then #25-4c = 0# and hence #c = 25/4#.