"Did our six-pump rocket ever reach 25 meters?" Comet asks, reading the launch log.
Wren writes -5t² + 20t = 25 and divides by -5. "t² - 4t + 5 = 0. Try the formula."
Comet works it out. "b² - 4ac is 16 minus 20. Negative four. I cannot take the square root of a negative."
"So there is no real solution," Wren says. "The rocket never got to 25. What do you notice at 20?"
"t² - 4t + 4 = 0. The discriminant is zero. One solution, t = 2. It touched 20 exactly once, at the top."
Nova projects the three equations in a column. "Would you like a hint? Look at the sign under the root each time."
"Positive, two times. Zero, once. Negative, never," Comet says. "The discriminant is the whole story."
In the quadratic formula, the number under the root is b² - 4ac. It is called the discriminant.
If it is positive, there are two real solutions. If it is zero, there is one. If it is negative, there are none.
For the rocket, that means the height is reached twice, once at the top, or never.
Every row is the crew's own made-up launch. Each equation is the model set equal to the height, divided by -5.
| Flight time T | Height asked | Equation | Discriminant | Reached |
|---|---|---|---|---|
| 4 s | 15 m | t² - 4t + 3 = 0 | 4 | twice |
| 4 s | 20 m | t² - 4t + 4 = 0 | 0 | once |
| 4 s | 25 m | t² - 4t + 5 = 0 | -4 | never |
| 6 s | 40 m | t² - 6t + 8 = 0 | ? | ? |
| 6 s | 45 m | t² - 6t + 9 = 0 | ? | ? |
| 3 s | 10 m | t² - 3t + 2 = 0 | ? | ? |
| Statement | True or false? |
|---|---|
| A positive discriminant means two real solutions. | ? |
| 4 × 4 - 4 × 1 × 5 = -4 | ? |
| 4 × 4 - 4 × 1 × 4 = 0 | ? |
| A discriminant of zero means the height is reached twice. | ? |
| Dividing every term by -5 keeps the same solutions. | ? |
Sharp reading of the numbers. Tomorrow you write quadratic equations from Loft builds and decide which solutions make sense.