Comet unrolls a sheet of cardboard. "The launch pad. It needs to be 3 meters longer than it is wide, with 40 square meters."
"Call the width x," Wren says. "Then the length is x + 3, and the area is x(x + 3). What do you notice?"
"x² + 3x = 40. A quadratic equation." Comet moves the 40 over and factors. "(x + 8)(x - 5) = 0."
"So x is -8 or 5," Wren says. "Which one is a width?"
"Five. A pad cannot be negative eight meters wide." Comet measures five meters of cardboard.
Nova hovers over the sketch. "Would you like a hint? Always check both numbers in the story, not just the algebra."
"Width 5, length 8, area 40," Comet says. "It fits."
| Loft problem | Equation | Solutions | Makes sense |
|---|---|---|---|
| A pad 3 m longer than wide with area 40 m². Width x. | x² + 3x - 40 = 0 | x = -8 or 5 | x = 5 |
| Two tile counts differ by 2 and multiply to 48. Smaller count n. | n² + 2n - 48 = 0 | n = -8 or 6 | n = 6 |
| The 4-second rocket is at 15 m. Time t. | t² - 4t + 3 = 0 | t = 1 or 3 | both, up and down |
A negative length or a negative time is a true solution of the equation but not of the build. Set it aside and say why.
The rocket is the same: -5t² + 20t = 0 has t = 0 and t = 4. Only the second is a landing.
| Statement | True or false? |
|---|---|
| A quadratic equation can be written as ax² + bx + c = 0 with a not zero. | ? |
| The zero-product rule needs a zero on one side of the equation. | ? |
| Completing the square on x² + 8x adds 16. | ? |
| A negative discriminant means two real solutions. | ? |
| A negative solution for a width is kept because the algebra says so. | ? |
Terrific week. You can write and solve any quadratic equation four ways. Tomorrow is Loft Day.