Step by step
The question
Question: write the quadratic equation with roots −1 and −6.
roots −1 and −6
1 Sum and product
The sum of the roots: −1 plus −6 is −7.
sum = −1 + (−6) = −7
The product: −1 times −6 is 6.
product = −1 × (−6) = 6
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 − (−7)x + 6 = 0
Careful!
The sum goes in with a minus in front: minus −7.
Careful: not x² − 7x + 6 = 0 ✗
Tidy the signs: x² + 7x + 6 = 0.
x² + 7x + 6 = 0
3 Check
Let's check: put each root in.
x = −1 gives 0.
x = −1: (−1)² + 7(−1) + 6 = 0
And x = −6 gives 0 too.
So the equation is right!
x = −6: (−6)² + 7(−6) + 6 = 0
Let's look back.
First: sum and product.
Then: write the equation.
Last: check.
Answer
x² + 7x + 6 = 0