find quadratic equation with a root double the other

certainly! let's go through how to find a quadratic equation where one root is double the other root. we'll derive the equation step-by-step and provide a code example in python.

### understanding quadratic equations

a quadratic equation is generally expressed in the form:

\[ ax^2 + bx + c = 0 \]

where:
- \( a \), \( b \), and \( c \) are coefficients.
- the roots of the equation can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### problem statement

let’s say we have two roots: \( r_1 \) and \( r_2 \), where \( r_2 = 2r_1 \).

### step 1: define the roots

let’s denote:
- \( r_1 = r \)
- \( r_2 = 2r \)

### step 2: relationship between roots and coefficients

from vieta's formulas:
1. sum of roots: \( r_1 + r_2 = -\frac{b}{a} \)
2. product of roots: \( r_1 \cdot r_2 = \frac{c}{a} \)

substituting our roots:
1. \( r + 2r = -\frac{b}{a} \) → \( 3r = -\frac{b}{a} \)
2. \( r \cdot 2r = \frac{c}{a} \) → \( 2r^2 = \frac{c}{a} \)

### step 3: choosing coefficient \( a \)

for simplicity, let’s choose \( a = 1 \). thus:
- \( b = -3r \)
- \( c = 2r^2 \)

### step 4: form the quadratic equation

the quadratic equation becomes:

\[ x^2 - 3rx + 2r^2 = 0 \]

### step 5: implementation in code

now, let's implement this in python. we will create a function that takes a root \( r \) and returns the coefficients of the quadratic equation.

### step 6: example calculation

if we take \( r_1 = 2 \), then:

- \( r_2 = 2 \cdot 2 = 4 \)
- the coefficients will be:
- \( a = 1 \)
- \( b = -3 \cdot 2 = -6 \)
- \( c = 2 \cdot 2^2 = 8 \)

thus, the equation will be:

\[ 1x^2 - 6x + 8 = 0 \]

### conclusion

this tutorial has shown how to derive a quadratic equation given that one root is double the other, and provided a simple python implementation to compute the coefficients of the equation. you can modify the root value and see how the equation changes accordingly!

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