"The formula gave us 2 ± i," Comet says, tapping yesterday's page. "Can we get there another way?"
"Complete the square," Wren says. "x² - 4x + 5 = 0. Half of -4 is -2, and (-2)² is 4."
He writes (x - 2)² + 1 = 0. "So (x - 2)² = -1. What do you notice?"
"A square equal to negative one," Comet says. "That is i squared. So x - 2 is i or -i."
Nova dims to a hint. "Would you like a hint? Add 2 to both sides, both times."
"x = 2 + i or x = 2 - i," Comet says. "The same pair, no formula needed."
"Two roads, one answer," Wren says. "Now let us try the heights in the log, one by one."
Completing the square turns x² - 4x + 5 = 0 into a square equal to a number. Read each step, then put them in order below.
The quadratic formula is this same method done once for every equation. Both roads must agree.
Notice the pair: 2 + i and 2 - i. Complex solutions of a quadratic with real coefficients always come in such a pair.
Every row is the crew's own made-up question about the arc. Fill the blanks in your head, then check with the questions below.
| Height asked | Equation | Discriminant | Solutions |
|---|---|---|---|
| 3 | x² - 4x + 3 = 0 | 4 | x = 1 or x = 3 |
| 4 | x² - 4x + 4 = 0 | 0 | x = 2 |
| 5 | x² - 4x + 5 = 0 | ? | ? |
| 6 | x² - 4x + 6 = 0 | ? | ? |
| Statement | True or false? |
|---|---|
| x² - 4x + 5 = 0 and (x - 2)² = -1 have the same solutions. | ? |
| 4 × 4 - 4 × 1 × 6 = -8 | ? |
| If 2 + i solves a quadratic with real coefficients, so does 2 - i. | ? |
| A negative discriminant means the equation has no solutions at all. | ? |
| Completing the square and the quadratic formula can give different answers. | ? |
Careful reasoning. Tomorrow you use the week's tools on mixed problems and decide which answers make sense in the build.