# Thread: Finding the new equation

1. ## Finding the new equation

Question: Find an equation whose roots are the reciprocals of the roots of 2x^2 + x -4 = 0

What i did so far:

sum = -b/a = -1/2 --> reciprocal = -2
product = c/a = -2 --> reciprocal = -1/2

Therefore, x^2 - (sum)x + (product) = x^2 - (-2)x + (-1/2)
= 2x^2 + 4x -1

From other sources, another answered showed: 4x^2 + x -2

Can someone verify whether what i did is correct?

2. Originally Posted by onenameless
Question: Find an equation whose roots are the reciprocals of the roots of 2x^2 + x -4 = 0

What i did so far:

sum = -b/a = -1/2 --> reciprocal = -2
product = c/a = -2 --> reciprocal = -1/2

Therefore, x^2 - (sum)x + (product) = x^2 - (-2)x + (-1/2)
= 2x^2 + 4x -1

From other sources, another answered showed: 4x^2 + x -2

Can someone verify whether what i did is correct?
The sum of the roots in the new equation is not necessarily going to be the reciprocal of the sum of the roots in the old equation.

3. Originally Posted by icemanfan
The sum of the roots in the new equation is not necessarily going to be the reciprocal of the sum of the roots in the old equation.
But the question asks it to be the reciprocal of the old equation.

4. Originally Posted by onenameless
But the question asks it to be the reciprocal of the old equation.
Suppose you have an equation with roots 2 and 3. Then the sum of the roots is 5.

The reciprocals of 2 and 3 are 1/2 and 1/3, and their sum is 5/6, which is not the reciprocal of 5.

So the roots are the reciprocals but the sum of the roots is not a reciprocal.

I would suggest finding the roots of the original equation as it will make your job easier.

5. Originally Posted by icemanfan
Suppose you have an equation with roots 2 and 3. Then the sum of the roots is 5.

The reciprocals of 2 and 3 are 1/2 and 1/3, and their sum is 5/6, which is not the reciprocal of 5.

So the roots are the reciprocals but the sum of the roots is not a reciprocal.

I would suggest finding the roots of the original equation as it will make your job easier.
Oh right right, my mistake, i think this should be done like:

Normally, sum = a + b, but the reciprocal is:
sum = 1/a + 1/b
sum = (b + a) / ab

b + a is the sum, and sum = -b/a, and therefore is equal to: -1/2
ab is the product, and product = c/a, and therefore is equal to: -2

So sum = (-1/2) / (-2) = 1/4

Normally, product = ab, but the reciprocal is:
product = (1 / a)(1 / b)
= (1 / ab)
= (1 / -2)
Therefore the equation is x^2 + 1/4x - 1/2 = 0 OR 4x^2 - x - 2 = 0