Step by step
The question
Question: write the quadratic equation with roots −1/2 and −4.
roots −12 and −4
1 Sum and product
The sum of the roots: −1/2 plus −4 is −9/2.
sum = −12 + (−4) = −92
To add fractions, we use the same bottom, 2: as in the fractions lesson.
−12 + (−4) = −12 + (−82) = −92
The product: −1/2 times −4 is 2.
product = −12 × (−4) = 2
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 − (−92)x + 2 = 0
Careful!
The sum goes in with a minus in front: minus −9/2.
Careful: not 2x² − 9x + 4 = 0 ✗
Tidy the signs.
x² + 92x + 2 = 0
There are fractions.
We multiply every part by 2, on both sides.
0 times 2 is still 0.
× 2: no fractions
So the equation is 2x² + 9x + 4 = 0.
2x² + 9x + 4 = 0
3 Check
Let's check: put each root in.
x = −1/2 gives 0.
x = −12: 2(−12)² + 9(−12) + 4 = 0
And x = −4 gives 0 too.
So the equation is right!
x = −4: 2(−4)² + 9(−4) + 4 = 0
Let's look back.
First: sum and product.
Then: write the equation.
Last: check.
Answer
2x² + 9x + 4 = 0