Algebra 2 · Unit 4 · Lesson 15
Working Backward
Learning Target
I can find the real or imaginary part of a complex number when I know something about its product with another complex number.
🧑🏫 This lesson brings together multiplication of complex numbers (Lessons 12–14) and asks students to reverse the process. The core skill: given a product and one factor, use the multiplication pattern to set up and solve one linear equation for a missing part. No simultaneous equations — one unknown, one equation per problem.
Warm-Up · What's Missing?
10 min
Name the parts.
Real Part
a + bi
In a + bi, the real part is the number a.
Imaginary Part
a + bi
In a + bi, the imaginary part is the number b.
(10 + 4i) − ( + i) = C
Part A
If C is a real number, what goes in the blanks?
The imaginary blank must be 4. The real blank can be any number. Example: 3 + 4i
Part B
If C is an imaginary number, what goes in the blanks?
The real blank must be 10. The imaginary blank can be any number (≠ 4). Example: 10 + 9i
🧑🏫 Purpose: activate the vocabulary "real part" and "imaginary part" before the main activity. For Part A: imaginary parts must cancel → second blank = 4, first blank is free. For Part B: real parts must cancel → first blank = 10, second blank is free (but ≠ 4 or C would be 0, not imaginary). Cold-call 2–3 pairs.
Real parts combine with real parts.
Imaginary parts combine with imaginary parts.
When you add or subtract complex numbers, the real parts combine only with each other — and the imaginary parts combine only with each other. They never mix.
(10 + 4i) − (3 + 4i) = (10−3) + (4−4)i = 7 + 0i = 7
Big idea for today: This separation also happens when you multiply. Knowing which part equals what gives us an equation to solve — even if one part of a factor is unknown.
🧑🏫 Bridge to the activity: "Today we flip the question. Instead of multiplying two known complex numbers, you'll know the product and figure out a missing piece." Spend no more than 3 min here.
Activity · Launch
25 min total
What Part Is Missing?
When we multiply two complex numbers, the real part and imaginary part of the result always come from the same pattern:
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
↑ real part of the product
↑ imaginary part of the product
If one part of a factor is unknown, we can use what we know about the product to write one equation and solve for the unknown.
Work on the reference slide first, then try the problems.
🧑🏫 Keep this launch under 3 min. Show the formula, point to the color coding, then go directly to the How To slide. Students do not need to memorize the formula — they practiced distributing in Lessons 12–14. The How To slide walks through the process step by step.
How to find a missing part of a complex product
| Step |
What to do |
Example: (2 + i)(a + 2i) is real. Find a. |
| 1 |
Multiply — keep the unknown as a variable |
(2+i)(a+2i) = 2a + 4i + ai + 2i² = 2a−2 + (a+4)i |
| 2 |
Label the real part and the imaginary part |
Real part: 2a−2 Imaginary part: (a+4) |
| 3 |
Use what you know to write one equation |
"Is real" → imaginary part = 0 → a + 4 = 0 |
| 4 |
Solve for the unknown |
a = −4 |
| 5 |
Check — substitute back and verify |
(2+i)(−4+2i) = −8+4i−4i+2i² = −10 ✓ (real) |
🧑🏫 Leave this slide on screen during the activity problems. Students reference it independently. Walk through it one click at a time before releasing students to work. Step 3 is the key idea — students must identify which part of the product tells them what to set equal to.
What Part Is Missing?
Multiply first. Then use the real or imaginary part to write one equation.
Problem 1
(2 + i)(a + 2i) is a real number. Find a.
What must the imaginary part of the product equal?
Expand: (2+i)(a+2i) = 2a−2 + (a+4)i
"Is real" → a+4 = 0
a = −4 · Product = −10
Problem 2
The imaginary part of (3 + 2i)(1 + bi) is 8. Find b.
Expand the product. Which part gives you an equation?
Expand: (3+2i)(1+bi) = 3−2b + (3b+2)i
Imaginary part = 8 → 3b+2 = 8 → 3b = 6
b = 2 · Product = −1 + 8i
Explain
Ask
Change
I set ___ equal to ___ because ___.
Which part did you use to write the equation?
Can you check your answer by multiplying?
🧑🏫 Circulate. Watch for students who try to set up two equations — redirect to "we only have one unknown, so we only need one equation." Common error on P2: students use the real part (3−2b) instead of the imaginary part. Ask: "What information were you given about the product?"
What Part Is Missing?
Choose which part to use. You decide.
Problem 3
The real part of (1 + 3i)(c + 4i) is −10. Find c.
Expand the product. Write an equation for the real part. Solve.
Expand: (1+3i)(c+4i) = c+4i+3ci+12i² = (c−12) + (3c+4)i
Real part = −10 → c−12 = −10 → c = 2
c = 2 · Product = −10 + 10i
Notice: After you find c = 2, you also know the imaginary part of the product without any extra work — it's 3(2) + 4 = 10. The formula gives you both parts at once.
Explain
Ask
Change
I set ___ equal to ___ because ___.
How did you decide which part to use?
Can you find the imaginary part of the product too?
🧑🏫 This problem is slightly harder because the real part condition involves a subtraction (c − 12). Watch for sign errors. The "Notice" callout extends the thinking: once c is found, both parts of the product are fully determined. This bridges toward Lesson 16 (quadratic equations with complex solutions).
What was the strategy?
The key move: When you know something about the product (it's real, its real part is some value, its imaginary part is some value), translate that into an equation for the unknown — then solve.
If the product is real…
set the imaginary part = 0
If the product is imaginary…
set the real part = 0
Discussion: "What information did you need before you could write an equation? What would have made the problem impossible to solve?"
🧑🏫 Invite 2–3 students to explain their process on one problem. Highlight that real and imaginary parts always combine separately — which is why one equation is enough when there is only one unknown. Ask: "Could you have solved these if you had two unknowns?" (no — you'd need more information).
Working Backward
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
Every product of two complex numbers has a real part and an imaginary part that come from combining the parts of the original numbers in a fixed way. Because of this, we can work backward: use information about the product to find an unknown part of a factor.
When you work backward:
1. Multiply — keeping the unknown as a variable
2. Use the given condition to write one equation
3. Solve that equation
🧑🏫 Close with: "Tomorrow we'll see this idea connect to something much bigger — solving equations that have complex number solutions." 2 min max.
How to find a missing part of a complex product
| Step |
What to do |
Example: (2 + i)(a + 2i) is real. Find a. |
| 1 |
Multiply — keep the unknown as a variable |
(2+i)(a+2i) = 2a−2 + (a+4)i |
| 2 |
Label the real part and the imaginary part |
Real: 2a−2 Imaginary: (a+4) |
| 3 |
Use what you know to write one equation |
"Is real" → imaginary part = 0 → a+4 = 0 |
| 4 |
Solve for the unknown |
a = −4 |
| 5 |
Check — substitute back and verify |
(2+i)(−4+2i) = −8+4i−4i+2i² = −10 ✓ (real) |
🧑🏫 Leave this slide on screen during the cool-down. Students reference it independently.
How Do You Know?
The expression below is a real number.
(4 + 2i)(a + 2i)
Find the value of a. Show your work.
Write your answer in the form a = ___
🧑🏫 Answer: a = −4
Work: (4+2i)(a+2i) = 4a−4 + (2a+8)i
"Is real" → 2a+8 = 0 → a = −4 · Product = −20
Common error: Setting the real part 4a−4 = 0 (gives a = 1, not correct). Ask: "What part has to be zero for a complex number to be real?"
If struggling: Direct to the How To reference (slide 10).
Amplify Activity Builder — CL
Math Response component name: answer
Correct answer: −4
content:
when answer.submitted and answer.latex = "-4"
"Correct! When a = −4, the imaginary part is 2(−4) + 8 = 0."
when answer.submitted and answer.latex = "- 4"
"Correct! When a = −4, the imaginary part is 2(−4) + 8 = 0."
when answer.submitted and answer.latex = "1"
"Almost — you set the real part equal to zero. Which part must be zero for the product to be real?"
when answer.submitted and answer.latex = "4"
"Check your signs — expand (4+2i)(a+2i) and identify the imaginary part."
when answer.submitted and answer.latex = "-8"
"I see what happened — check which coefficient goes with a in the imaginary part: 2a + 8 = 0."
when answer.submitted
"Expand first. Label the real part and imaginary part. Then set the imaginary part equal to 0 and solve."
otherwise ""