Step by step
The question
Question: write the quadratic equation with roots 2 and 4.
roots 2 and 4
1 Sum and product
The sum of the roots: 2 plus 4 is 6.
sum = 2 + 4 = 6
The product: 2 times 4 is 8.
product = 2 × 4 = 8
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 − 6x + 8 = 0
Careful!
The sum goes in with a minus in front: minus 6.
Careful: not x² + 6x + 8 = 0 ✗
Tidy the signs: x² − 6x + 8 = 0.
x² − 6x + 8 = 0
3 Check
Let's check: put each root in.
x = 2 gives 0.
x = 2: 2² − 6(2) + 8 = 0
And x = 4 gives 0 too.
So the equation is right!
x = 4: 4² − 6(4) + 8 = 0
Let's look back.
First: sum and product.
Then: write the equation.
Last: check.
Answer
x² − 6x + 8 = 0