Question

If A,B,C and D are nonzero numbers such that c and d are solutions

of x^2 + Ax + B =0 and x^2 + Cx + D = 0, find A + B + C + D.

i) -2 ii) -1 iii) 0 iv) 1 v) 2.

Provide intuitive answer.

Answer 1

i) -2

Explanation:

In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

c and d are solutions(roots) of
`x^2 + Ax + B=0`

a and b are solutions(roots) of
`x^2 + Cx + D = 0`

`c+d=-a` eq. 1

`cd=b` eq. 2

`a+b=-c` eq. 3

`ab=d` eq. 4

from eq. 1 we get
`d=-a-c`

from eq. 3 we get
`b=-a-c`

so
`d=b`

now substitute
`d=b` in eq. 2

`cb=b`

`c=b/b=1`

now substitute
`d=b` in eq. 4

`ab=d`

`ad=d`

`a=d/d=1`

now substitute the values of
`c=1 and a=1` in eq. 1

`d=-a-c=-1-1=-2`

similarly, substitute the values of
`c=1 and a=1` in eq. 3

`b=-a-c=-1-1=-2`

Finally,

`A+B+C+D= 1-2+1-2=-2`

Lets verify and see if our answer is right!

`x^2 + Ax + B =0`

Substituting the values of A and B

`X^2+x-2=0`

we know that c and d are solutions of this equation so they must satisfy the equation.

put
`c=1`

`(1)^2+1-2=0`

`2-2=0`

`0=0` (proved)

put
`d=-2`

`(-2)^2-2-2=0`

`4-4=0`

`0=0` (proved)