A grown-up unhooks the hanging planter and sets it on the bench. "Lab day," Comet says. "We measure before we build."
She sets a squeeze bottle at the end of the bench and squeezes. Water arcs across the wood and lands.
Wren runs the tape along the bench. "Reach, four units. Top, about four units high. Every reading goes in the log."
"So h = -x² + 4x," Comet says. "Zero at x = 0 and x = 4. Top at x = 2. What do you notice?"
"The planter base sits at 5," Wren says. "Yesterday we tried that. Today let us try 3 and 4."
Nova projects three equations in a column. "Would you like a hint? Watch the discriminant change as the height climbs."
"Two crossings at 3, one touch at 4, nothing at 5," Comet says. "The planter has to come down."
These readings are the crew's own, made up for the Glass House. Yours will differ, and that is the point of a lab.
| Reading | Value (tape units) | What it fixes |
|---|---|---|
| Reach R | 4 | h = 0 at x = 0 and x = 4 |
| Top T | 4 | a × 2 × 2 = 4, so a = -1 after the sign flip |
| Model | -x² + 4x | the crew's fit, not a law |
| Height H | Equation | Discriminant | Real solutions |
|---|---|---|---|
| 3 | x² - 4x + 3 = 0 | 4 | two real solutions |
| 4 | x² - 4x + 4 = 0 | 0 | one real solution |
| 5 | x² - 4x + 5 = 0 | -4 | no real solution |
At height 3 the arc crosses twice: x = 1 and x = 3. At height 4 it touches once, at the top.
At height 5 the solutions are x = 2 - 1i and x = 2 + 1i. Complex, so the water never arrives.
| Statement | True or false? |
|---|---|
| 0 - 2 × 2 + 4 × 2 = 4 | ? |
| 0 - 1 × 1 + 4 × 1 = 3 | ? |
| 4 × 4 - 4 × 1 × 3 = 4 | ? |
| A complex solution means the arc reaches that height at an imaginary distance along the bench. | ? |
| The model h = -x² + 4x is the crew's own fit to their readings. | ? |
Good measuring. Tomorrow you solve complex quadratics by completing the square and see why the formula works.