For equation ax^2 + bx + c = 0, find the condition such that one root may be (i) 4 times the other (ii) m times the other (iii) more than the other by k

The equation given is ax^2 + bx + c = 0

(i) One root is 4 times the other. We can assume them as t and 4t.

Sum of roots = -b/a
=> t + 4t = -b/a
=> 5t = -b/a
=> t = -b/5a               (equation 1)

Product of roots = c/a
=> t * 4t = c/a
=> 4t^2 = c/a

Substituting the value of t from equation 1, we'll get

=> 4(-b/5a)^2 = c/a
=> (4b^2)/(25a^2) = c/a
=> (4b^2)/25a = c
=> 4b^2 = 25ac


Hence, if 4b^2 = 25ac is true, one root of ax^2 + bx + c will be 4 times the other.

(ii) One root is m times the other. We can assume them as t and mt.

Sum of roots = -b/a
=> t + mt = -b/a
=> (m+1)t = -b/a
=> t = -b/a(m+1)               (equation 2)

Product of roots = c/a
=> t * mt = c/a
=> mt^2 = c/a


Substituting the value of t from equation 2, we'll get

=> m(-b/a(m+1))^2 = c/a
=> (mb^2)/a^2(m+1)^2 = c/a
=> (mb^2)/a(m+1)^2 = c
=> mb^2 = ac (m+1)^2

Hence, if mb^2 = ac (m+1)^2 is true, one root of ax^2 + bx + c will be m times the other.

(iii) One root is more than the other by k, which means the difference of roots will be k. We know that - 

Difference of roots = √D /a

where D is discriminant of the equation, which is equal to (b^2 - 4ac). We are given that the difference of roots is k. So,

=> k = √D /a
=> k = √(b^2 - 4ac) / a
=> ak = √(b^2 - 4ac)

Squaring both the sides, we'll get

=> (ak)^2 = b^2 - 4ac
=> a^2.k^2 = b^2 - 4ac

Hence, if a^2.k^2 = b^2 - 4ac is true, one root of ax^2 + bx + c will be more than the other by k.