Suppose the quadratic polynomial P ( x )=a x²+ b x + c has positive coefficients a , b , c in arithmetic progression in that order. If P ( x )=0 has integer roots and , then + + equals

Step-by-step solution

array l P ( x )= ax ²+ bx + c = a ( x - )( x - ) and + + +1-1=( +1)( +1)-1 = ( a - b + c ) a -1 + + = b a -1= ₁-1 array i.e., b a is integer = ₁ If b = a ₁ then, c = a (2 ₁-1 ) because a , b , c are in A.P. aligned P ( x ) &= ax ²+ a ₁ x + a (2 ₁-1 ) &= a [ x ²+ ₁ x + (2 ₁-1 ) ] aligned D = ₁²-4 (2 ₁-1 ) is perfect square for integral roots D = ₁ ²-8 ₁+4 is perfect square Let D = ( ₁-4 )²-12= k ² where k I ( ₁-4- k ) ( ₁-4+ k )=12 This gives ₁-4- k =2 array c ₁-4=4 & & k=1 ₁=8 + + =8-1=7 array