Question

If the roots of the equations 2x^2 - 3x + 5 = 0 and ax^2 + bx + 2 = 0 are reciprocal of the roots of the other then (a,b) equals to?
Answer is (5,-3)
Please explain how.

Answer 1

We have 2x2+7x+5=0

⟹x2+72x+52=0

Let the roots of above equation be p and q .Then,

Sum of roots = p+q = −72

Product of roots = pq=52

Given, roots of ax2+bx+c=0 or x2+bax+ca=0 are 1p and 1q.

⟹ sum of roots =−ba

=1p+1q

=p+qpq

=−7252

=−75

⟹ba=75

and

product of roots = ca

=1pq

=25

∴x2+bax+ca=0

⟹x2+75x+25=0

⟹5x2+7x+2=0

On comparing, with ax2+bx+c=0 , we get a=5 and c=2.

⟹a−c=3

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Let p and q are the roots of 2x^2+7x+5 = 0.

p+q = -7/2………….(1)

p.q = 5/2……….. ….(2)

Thus , 1/p and 1/q are the roots of ax^2+bx+c = 0

or x^2 + b/a.x+ c/a = 0

1/p+1/q = -b/a ………………(3)

1/p.1/q = c/a……………………(4).

from eq (3)

(q+p)/p.q = -b/a

(-7/2)/(5/2)= - b/a

or b/a=7/5…………..(5)

from eq. (4)

1/p.q = c/a

c/a = 2/5…………………(6)

The given eq. is x^2+(b/a).x+c/a =0

or x^2+(7/5).x+2/5 =0

or 5x^2+ 7x + 2 = 0…………….(7)

equating the given eq. ax^2+bx+c=0 to eq.(7)

a =5 , b = 7 and c = 2 .

And a-c = 5 - 2 =3

Answer 2

Step-by-step explanation:

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