Step by step
The question
Question: write the quadratic equation with roots 5 and −5.
roots 5 and −5
1 Sum and product
The sum of the roots: 5 plus −5 is 0.
sum = 5 + (−5) = 0
The product: 5 times −5 is −25.
product = 5 × (−5) = −25
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 − 0x + (−25) = 0
Careful!
The product goes in with its own sign.
Here it is −25.
Careful: not x² + 25 = 0 ✗
Tidy the signs: x² − 25 = 0.
x² − 25 = 0
3 Check
Let's check: put each root in.
x = 5 gives 0.
x = 5: 5² − 25 = 0
And x = −5 gives 0 too.
So the equation is right!
x = −5: (−5)² − 25 = 0
Let's look back.
First: sum and product.
Then: write the equation.
Last: check.
Answer
x² − 25 = 0