Quadratic Functions and Equations in One Variable
How to Form a quadratic from its roots
Use this to build the quadratic equation when you are told its two roots.
Before you start
- Adding and multiplying positive and negative numbers
- The idea that a root makes a bracket zero
- Clearing fractions by multiplying through
When to use it
Use this to build the quadratic equation when you are told its two roots.
The steps
- Call the two roots α and β.
- Find their sum α + β and product αβ.
- Use the form x² − (sum)x + (product) = 0.
- Substitute the sum and product you found.
- Multiply through to clear fractions if needed.
Worked example
Form a quadratic equation whose roots are 2 and 5.
- Call the two roots α = 2 and β = 5.
- Find their sum α + β = 2 + 5 = 7 and their product αβ = 2 × 5 = 10.
- Use the form x² − (sum)x + (product) = 0.
- Substitute the sum and product: x² − 7x + 10 = 0.
- There are no fractions, so nothing needs to be multiplied through.
A second example, with a twist
One root is a fraction, so the sum and product are fractions and you must multiply through to clear them. Form a quadratic equation whose roots are ½ and −3.
- Call the two roots α = ½ and β = −3.
- Find their sum α + β = ½ + (−3) = −5/2 and their product αβ = ½ × (−3) = −3/2.
- Use the form x² − (sum)x + (product) = 0.
- Substitute: x² − (−5/2)x + (−3/2) = 0, which is x² + (5/2)x − 3/2 = 0.
- Multiply through by 2 to clear the fractions: 2x² + 5x − 3 = 0.
Practise this in a KBAT problem
Frequently asked questions
Why is it minus the sum but plus the product?
It comes from expanding (x − α)(x − β). Multiplying out gives x² − (α + β)x + αβ, so the sum appears with a minus sign and the product with a plus.
Memorise the pattern x² − (sum)x + (product) = 0 and the signs will always be right.
Do I have to multiply through to clear fractions?
You do not have to, but it makes a tidier answer with whole-number coefficients, which is what exams usually expect. Multiplying every term by the same number keeps the equation balanced and the roots unchanged.
Choose the smallest number that removes all the denominators at once.
Can I just expand (x − α)(x − β) instead?
Yes, that gives exactly the same equation. Writing (x − 2)(x − 5) and multiplying out lands you at x² − 7x + 10 = 0.
The sum-and-product method is just a quicker route to the same result, and it is handy when the roots are awkward numbers.