Find Integral values of x for which x2 + 19x + 46 is a perfect square?

 x2+19x+46

=x2+18x+81+x-35

=(x+9)2+(x-35)

for the above equation to become perfect square x-35 must be equal to zero

therefore x=35

similarly, x2+19x+46=x2+20x+100-(x+54)

=(x+10)2-(x+54)

here x+54=0

=>  x=-54

x2+19x+46=x2+22x+121-3(x+25)

=(x+11)2-3(x+25)

here  x+25=0

=>  x=-25

x2+19x+46=x2+16x+64+3x-18

=(x+8)2+3(x-6)

here  x-6=0

=>  x=6

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