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Algebra 2 9-12 / Week 01 / Friday
5/6
Week 01 · Complex Numbers and the Arc That Never Lands

Friday

Which solution makes sense?
// When the square root of a negative shows up
⏱ about 20 min

Friday: Which Solution Makes Sense?

Comet lowers the planter hook two links. "New height, 3 units. Now the arc should cross it twice."

Wren checks the log. "x² - 4x + 3 = 0. Solutions 1 and 3. The water passes height 3 going up and coming down."

"So we hang the planter at distance 1 or distance 3," Comet says. "Three keeps it clear of the bench edge."

"And the old height of 5 gave 2 ± i," Wren says. "What do you notice about how we read each answer?"

"Real solutions are places on the bench. Complex solutions mean the arc never gets there."

Nova projects both arcs side by side. "Would you like a hint? Keep both kinds in the log. Both are true."

"Both true," Comet says, "but only one gets the planter watered."

Reading a solution

Every quadratic has two solutions when you count complex ones. The build decides which to use.

A real solution is a distance, a time or a height the crew can measure. A complex solution says the thing never happens.

A negative real solution is real but may still be set aside, like a distance behind the sprinkler.

Question in the buildEquationSolutionsRead in the build
Where is the arc at height 3?x² - 4x + 3 = 0x = 1 or 3two places on the bench
Where is the arc at its top?x² - 4x + 4 = 0x = 2one place, the top
Where is the arc at height 5?x² - 4x + 5 = 0x = 2 - 1i or 2 + 1inever

The week in one table

SkillRuleExample
the imaginary uniti² = -1(3i)² = -9
addingreal with real, i with i(2 + i) + (1 - 3i) = 3 - 2i
multiplyinglike binomials, then i² = -1(2 + i)(1 - 3i) = 5 - 5i
the discriminantb² - 4acfor x² - 4x + 5: -4
complex solutionsformula or complete the squarex = 2 - 1i or 2 + 1i
MIXED REVIEW
  • Read the question.
  • Tap your answer.
What is (2 + i) + (1 - 3i)?
What is (2 + i)(1 - 3i)?
What is i¹³?
What are the solutions of x² - 2x + 2 = 0?
WHICH ANSWER FITS THE BUILD?
  • Read the question.
  • Tap your answer.
The arc model gives x = 1 or x = 3 for height 3. Which word describes these solutions?
For height 5 the solutions are 2 + i and 2 - i. What does the crew conclude?
A time equation gives t = 3 or t = -1. Which solution does the crew keep?
The discriminant of x² - 4x + 6 is ____. Type the number.
How many real solutions does an equation with a negative discriminant have? Type the number.
In a + bi, the letter b is called the ____ part. Type one word.
StatementTrue or false?
Every quadratic equation has two solutions if complex ones are counted.?
(1 - 2i)(1 + 2i) = 5?
0 - 3 × 3 + 4 × 3 = 3?
A complex solution tells the crew where on the bench to hang the planter.?
i⁴ = 1.?
WHY THIS EXERCISEDeciding which answer fits the build is the last step of every problem this course asks.
Try it
Write x² - 4x + c = 0 on a card and try c = 1, 4 and 7. Work out each discriminant.
Which c gives real places on the bench, which gives the top, and which gives a complex pair?
Draw the arc with the planter at height 3 and again at height 5. Mark where the water crosses each.

A full week. Tomorrow is Garden Day: you show your family an arc of your own and the number that never lands.

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