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Ganesh and Sarath were given a quadratic equation in x to solve. Ganesh made a mistake in copying the constant term of the equation and got a root as 12. Sarath made a mistake in copying the coefficient of x as well as the constant term and got a root as 2. Later they realized that the mistakes they committed were only in copying the signs. The difference between the roots of the original equation is

- 2
- 10
- 4
- Cannot be determined.

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Best answer

Option 1 is correct, i.e. difference between the roots is 2.

Suppose the original equation is ax^2 + bx + c, then

1. The equation copied by Ganesh is ax^2 + bx – c, and one root of this equation is 12.

2. The equation copied by Sarath is ax^2 - bx – c, and one root of this equation is 2.

From point 1, it is clear that a(12)^2 + 12b – c = 0,

144a + 12b – c = 0 ------------------------ equation 1

From point 2, it is clear that a(2)^2 - 2b – c = 0,

4a - 2b – c = 0 -------------------------- equation 2

On solving equations 1 & 2, it can be seen that :

b = -10a,

c = 24a,

What we have to calculate?

We have to calculate the difference between the roots of the equation: ax^2 + bx + c,

i.e. difference =

((-b + square root (b^2 – 4ac))/2a) -

((-b - square root (b^2 – 4ac))/2a)

which on simplification gives difference = (square root (b^2 – 4ac))/a.

putting the values of b & c in terms of a, in the above equation we get:

difference =

(square root ((-10a)^2 – 4a(24a)))/a

which will give difference = 2.

Suppose the original equation is ax^2 + bx + c, then

1. The equation copied by Ganesh is ax^2 + bx – c, and one root of this equation is 12.

2. The equation copied by Sarath is ax^2 - bx – c, and one root of this equation is 2.

From point 1, it is clear that a(12)^2 + 12b – c = 0,

144a + 12b – c = 0 ------------------------ equation 1

From point 2, it is clear that a(2)^2 - 2b – c = 0,

4a - 2b – c = 0 -------------------------- equation 2

On solving equations 1 & 2, it can be seen that :

b = -10a,

c = 24a,

What we have to calculate?

We have to calculate the difference between the roots of the equation: ax^2 + bx + c,

i.e. difference =

((-b + square root (b^2 – 4ac))/2a) -

((-b - square root (b^2 – 4ac))/2a)

which on simplification gives difference = (square root (b^2 – 4ac))/a.

putting the values of b & c in terms of a, in the above equation we get:

difference =

(square root ((-10a)^2 – 4a(24a)))/a

which will give difference = 2.

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