Math, asked by sandrasunil87, 1 day ago

Find the conditions that one root of ax^2+bx+c=0 may be
i) three times the other
ii) n times the other
iii) more than the other by h

Answers

Answered by primarycskts
0

Answer:

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Answered by pp209325
1

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.

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