Quadratic equations: the formula and the parabola
Change b and c in x² + bx + c = 0. Watch the parabola y = x² + bx + c move, find its roots with the formula, and see from the discriminant whether there are two roots, one or none.
Core concept
The roots of x² + bx + c = 0 are x = (−b ± √(b² − 4c))/2; the discriminant b² − 4c decides whether the parabola meets the x-axis twice, touches it once or misses it
Topic: Quadratic equations
A quadratic equation has x² as its highest power, like x² − 5x + 6 = 0. One way to solve it is to factorise: x² − 5x + 6 = (x − 2)(x − 3), so x = 2 or x = 3. Another is completing the square: x² + bx = −c becomes (x + b/2)² = b²/4 − c. Doing that in general gives the quadratic formula x = (−b ± √(b² − 4c))/2 for x² + bx + c = 0. The number under the root, D = b² − 4c, is the discriminant. If D > 0 there are two different roots; if D = 0 the two roots are equal; if D < 0 there is no real root, because a negative number has no real square root. On a graph, the roots are where the parabola y = x² + bx + c meets the x-axis.
What you will learn
- Solve x² + bx + c = 0 by factorisation and by the quadratic formula.
- Use the discriminant to tell whether the roots are real and different, equal or not real.
- Link the roots of an equation to where its parabola meets the x-axis.
Before you start
- Know how to factorise expressions like x² + bx + c and how to find square roots.
Materials
- Graph paper, a pencil and a ruler
Equipment
- A calculator for square roots
Procedure
- Record x² − 5x + 6 = 0.
- Record x² + 4x + 4 = 0.
- Record x² + 2x + 6 = 0.
- Record x² − 4 = 0 and x² + 2x − 12 = 0.
How this simulation is simplified
- The coefficient of x² is 1, so the equation is x² + bx + c = 0.