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Amit and Alok attempted to solve a quadratic equation. Amit made a mistake in writing down the constant term and ended up with roots (4, 3). Alok made a mistake in writing down coefficient of x to get roots (3, 2). The correct roots of the equation are:
Let, the original equation be $$ax^2+bx+c=0$$
So, sum of original roots will be -b/a and product of original roots will be c/a.
Now in the first case, Amit made a mistake in writing down the constant term.
Let, say in that case, the equation he wrote be $$ax^2+bx+d=0$$
Roots of this equation are given as (4,3)
So, sum of roots = -b/a = 4+3 = 7
or, b =-7a------>(1)
Now, let's pick the second case.
In the second case, he made a mistake in writing down the coefficient of x.
Let, say in that case, the equation he wrote be $$ax^2+ex+c=0$$
Roots of this equation are given as (3,2)
So, product of this roots = c/a = 3*2 = 6
or, c = 6a------>(2)
Putting the values of b and c in the original equation,
We get the equation as:
$$ax^2-7ax+6a=0$$
or, $$a\left(x^2-7x+6\right)=0$$
Now in a quadratic equation, we know, $$a\ne\ 0$$
So, the original equation is $$x^2-7x+6=0$$
And $$x^2-7x+6=\left(x-1\right)\left(x-6\right)=0$$
So, roots of this equation are 1 and 6
Option (B) 6,1 is the correct answer
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