Zeroes and coefficients of a quadratic
Change a, b and c in ax² + bx + c and watch where the parabola meets the x-axis. Check that the two zeroes always add up to −b/a and multiply to c/a.
Core concept
For ax² + bx + c with zeroes α and β, α + β = −b/a and αβ = c/a; when the discriminant b² − 4ac is negative there are no real zeroes
Topic: Polynomials
The zeroes of a polynomial are the values of x that make it 0; on the graph they are where the curve meets the x-axis. A quadratic ax² + bx + c can have two zeroes, α and β. Since ax² + bx + c = a(x − α)(x − β), multiplying out and comparing shows that the sum of the zeroes is α + β = −b/a and their product is αβ = c/a. For x² − 5x + 6 the zeroes are 3 and 2: 3 + 2 = 5 = −(−5)/1 and 3 × 2 = 6 = 6/1. When the discriminant b² − 4ac is negative, the parabola misses the x-axis and there are no real zeroes.
What you will learn
- Find the zeroes of a quadratic polynomial and show them on its graph.
- Check that α + β = −b/a and αβ = c/a.
- Use the discriminant to tell whether there are real zeroes.
- Write a quadratic from the sum and product of its zeroes.
Before you start
- Know how to factorise a quadratic and use the quadratic formula.
Materials
- Graph paper (optional)
Equipment
- A calculator
Procedure
- Record x² − 5x + 6.
- Change c to 2, then a to 2, and record 2x² − 5x + 2.
- Change b to 3 and record a quadratic with no real zeroes.
How this simulation is simplified
- The zeroes are rounded to two decimal places.