Question

Find the quadratic equation for roots 1/3 and -1/4

Answer 1

Here's the process.
If your roots are... say, 1 and 2, then the equations for these roots are
x=1 and x=2. In order to get a quadratic equation, all numbers and variables must be on the same side of the equation. So in both cases, subtract the number over to the left.
x-1=0 and x-2=0
Both of these are now your zeroes. These now become your factors for your quadratic equation.
(x-1)(x-2)=0
Then you multiply them out using the FOIL method:
Multiply the front numbers together, the outside numbers together, the inside numbers together, and the last numbers together.
x times x, x times -2, -1 times x, -1 times -2.
then you add them together.
x^2+(-2x)+(-x)+2=0
This comes out to be x^2-3x+2=0
And that is the equation.

Answer 2

quadratic equation for roots x1,x2
is
(x-x1)*(x-x2)=0
like this
quadratic equation for roots 2,4
(x-2)(x-4)=x^2-6x+8

Answer 3

Any polynomial with rational coefficients and roots \(r_1,r_2,...,r_n\) can be written in factored form as \[(x-r_1)(x-r_2)...(x-r_n)\]which is why you can do what the other posters suggest. The product of those terms will be 0 at every point where \(x\) is equal to one of the roots, and nowhere else.