Step by step
The question
Question: write the quadratic equation with roots 3 and −5.
roots 3 and −5
1 Sum and product
The sum of the roots: 3 plus −5 is −2.
sum = 3 + (−5) = −2
The product: 3 times −5 is −15.
product = 3 × (−5) = −15
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 − (−2)x + (−15) = 0
Careful!
The sum goes in with a minus in front: minus −2.
Careful: not x² − 2x − 15 = 0 ✗
Tidy the signs: x² + 2x − 15 = 0.
x² + 2x − 15 = 0
3 Check
Let's check: put each root in.
x = 3 gives 0.
x = 3: 3² + 2(3) − 15 = 0
And x = −5 gives 0 too.
So the equation is right!
x = −5: (−5)² + 2(−5) − 15 = 0
Let's look back.
First: sum and product.
Then: write the equation.
Last: check.
Answer
x² + 2x − 15 = 0