Two, one or none
The discriminant tells how many real roots there are.
Lift the curve
Look at x² − 4x + c = 0. Its curve is y = x² − 4x + c. Raising c lifts the whole curve upwards.
As c grows from 1 up to 6, the curve is lifted. How will the number of real roots change?
What this lesson covers
The idea
ax² + bx + c = 0 has two distinct real roots if b² – 4ac > 0, two equal real roots, both −b/2a, if b² – 4ac = 0, and no real roots if b² – 4ac < 0.
Lift the curve
Look at x² − 4x + c = 0. Its curve is y = x² − 4x + c. Raising c lifts the whole curve upwards.
As c grows from 1 up to 6, the curve is lifted. How will the number of real roots change?
- It stays at two
- It goes from two, to one, to none
- It grows: two, then three, then four
Lift the curve, read D
Step c up and down. Watch D = b² − 4ac, the verdict, and where the curve meets the x-axis. Find a c with D > 0, one with D = 0, and one with D < 0.
Three cases
ax² + bx + c = 0 has two distinct real roots if b² − 4ac > 0, two equal real roots, both −b/2a, if b² − 4ac = 0, no real roots if b² − 4ac < 0.
In the formula, D = 0 gives x = (−b ± 0)/2a, so both roots are −b/2a. On the graph, the curve just *touches* the x-axis at that one point. For x² − 4x + 4 = 0 it touches at x = 2.
D > 0: the curve cuts the x-axis twice. D < 0: the curve floats above (or below) the x-axis and never meets it.
Notes
D > 0: two distinct real roots. D = 0: two equal real roots, both −b/2a. D < 0: no real roots.
Check yourself
The equation 3x² − 5x + 2 = 0 has D = 25 − 24 = 1. What are its roots like?
x² − 6x + 9 = 0 has D = 0, so both roots equal −b/2a. What is the root?
Answer: 3
−b/2a = 6/2 = 3. Check: 9 − 18 + 9 = 0 ✓. The root 3 comes twice.
For which value of c does x² − 4x + c = 0 have two equal real roots?
Answer: 4
D = (−4)² − 4 × 1 × c = 16 − 4c. D = 0 gives c = 4. Then x² − 4x + 4 = (x − 2)² and the root is 2, twice.
x² + x + 1 = 0 has D = 1 − 4 = −3. How many real roots does it have?
- Two distinct real roots — correct. Yes! D > 0. (The roots are 1 and 2/3.)
- Two equal real roots. Equal roots need D = 0. Here D = 1.
- No real roots. No real roots need D < 0. Here D = 1.
- None — correct. Yes! D < 0 means no real roots: the curve never meets the x-axis.
- One. One (repeated) root needs D = 0. Here D = −3.
- Two. Two roots need D > 0. Here D = −3.