Step by step
The question
Question: write the quadratic equation with roots 1/3 and −2.
roots 13 and −2
1 Sum and product
The sum of the roots: 1/3 plus −2 is −5/3.
sum = 13 + (−2) = −53
To add fractions, we use the same bottom, 3: as in the fractions lesson.
13 + (−2) = 13 + (−63) = −53
The product: 1/3 times −2 is −2/3.
product = 13 × (−2) = −23
2 Write the equation
The equation with roots r and s is x minus r, times x minus s.
Opened up: x squared, minus the sum times x, plus the product.
x² − (sum)x + product = 0
(x − r)(x − s) = x² − (r + s)x + rs
Put the sum and the product in.
x² − (−53)x + (−23) = 0
Careful!
The sum goes in with a minus in front: minus −5/3.
Careful: not 3x² − 5x − 2 = 0 ✗
Tidy the signs.
x² + 53x − 23 = 0
There are fractions.
We multiply every part by 3, on both sides.
0 times 3 is still 0.
× 3: no fractions
So the equation is 3x² + 5x − 2 = 0.
3x² + 5x − 2 = 0
3 Check
Let's check: put each root in.
x = 1/3 gives 0.
x = 13: 3(13)² + 5(13) − 2 = 0
And x = −2 gives 0 too.
So the equation is right!
x = −2: 3(−2)² + 5(−2) − 2 = 0
Let's look back.
First: sum and product.
Then: write the equation.
Last: check.
Answer
3x² + 5x − 2 = 0