Step by step
The question
Question: write the quadratic equation with roots −3 and 8.
roots −3 and 8
1 Sum and product
The sum of the roots: −3 plus 8 is 5.
sum = −3 + 8 = 5
The product: −3 times 8 is −24.
product = −3 × 8 = −24
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 − 5x + (−24) = 0
Careful!
The sum goes in with a minus in front: minus 5.
Careful: not x² + 5x − 24 = 0 ✗
Tidy the signs: x² − 5x − 24 = 0.
x² − 5x − 24 = 0
3 Check
Let's check: put each root in.
x = −3 gives 0.
x = −3: (−3)² − 5(−3) − 24 = 0
And x = 8 gives 0 too.
So the equation is right!
x = 8: 8² − 5(8) − 24 = 0
Let's look back.
First: sum and product.
Then: write the equation.
Last: check.
Answer
x² − 5x − 24 = 0