If the roots of the quadratic equation a x 2 + b x + c = 0 are k + 1 k and k + 2 k + 1 , then the value of ( a + b + c ) 2 is equal to

Step-by-step solution

We have k + 1 k + k + 2 k + 1 = - b a . . . . ( i ) and k + 1 k · k + 2 k + 1 = c a ⇒ k + 2 k = c a or 2 k = c a - 1 = c - a a or k = 2 a c - a .....(ii) Now eliminate k putting the value of k in 1 st  relation, we get c + a 2 a + 2 c c + a = - b a ⇒ ( c + a ) 2 + 4 a c = - 2 b ( a + c ) ⇒ ( a + c ) 2 + 2 b ( a + c ) = - 4 a c Adding b 2 on both sides, ( a + b + c ) 2 = b 2 - 4 a c