LEARNER JOURNEY Solve (x − 3)(x + 2) = 0
Name the goalThe equation is a product equal to zero.
Choose the next moveSet x − 3 = 0 or x + 2 = 0.
Carry out the mathematicsx = 3 or x = −2.
Check and interpretEach value makes one factor zero, so the product is zero.
Start here
Build the concept from the beginning
Starting point
No prior equation-solving skill is assumed. Arithmetic operations, variables, equality, and every new algebra decision are explained on this page.
Learning target
Solve a factored quadratic by setting each factor equal to zero.
Evidence of mastery
Construct the reasoning path, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- equation
- A statement that two mathematical expressions have the same value.
- solution
- A value that makes an equation true when substituted for the variable.
- factor
- A quantity multiplied by another quantity to form a product.
- root
- A solution of an equation; for a polynomial, it makes the polynomial equal zero.
Look first, then name the mathematics
Build the visual meaning
- Product = 0
- First factor = 0
- Second factor = 0
- Roots
Read the complete visual relationship as text
- Product = 0
- Factor A = 0
- Factor B = 0
- Root 1
- Root 2
Definition in plain language
If a product equals zero, at least one factor must equal zero. For AB = 0, solve A = 0 or B = 0.
Why this matters
Factoring turns one quadratic equation into two simpler linear equations.
Reasoning path
See the method as a sequence
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Confirm the product equals zero.
Identify each complete factor.
Set each factor equal to zero.
Solve both linear equations and check both roots.
The same method in words
- Confirm the product equals zero.
- Identify each complete factor.
- Set each factor equal to zero.
- Solve both linear equations and check both roots.
Interactive decision model
Build the reasoning, one decision at a time
Solve (x + 6)(x − 2) = 0.
Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.
The reasoning path changes as each decision is selected.
Choose every step, then check the construction.
Model before practice
Three fully explained examples
Every example identifies the goal, explains the next move, carries out the algebra, and checks or interprets the result.
Solve x(x − 5) = 0
Name the goalThe first factor is x itself.
Choose the next moveSet x = 0 or x − 5 = 0.
Carry out the mathematicsx = 0 or x = 5.
Check and interpretDo not divide by x; that would lose the zero root.
Solve (2x + 1)(x − 4) = 0
Name the goalEach complete binomial is a factor.
Choose the next moveSolve 2x + 1 = 0 and x − 4 = 0.
Carry out the mathematicsx = −1/2 or x = 4.
Check and interpretBoth roots make the original product zero.
Practice with decreasing support
Guided practice
Practice immediately after the model
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. Solve (x − (2))(x − (5)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve (x − (-3))(x − (4)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Solve (x − (1))(x − (-6)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Independent practice
Six problems for repetition
Solve all six. Use the specific feedback to revise a method or sign error.
1. Solve (x − (-2))(x − (-7)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve (x − (0))(x − (8)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve (x − (3))(x − (9)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve (x − (-5))(x − (2)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve (x − (4))(x − (-1)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
6. Solve (x − (-6))(x − (-2)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. Solve (x − (7))(x − (1)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve (x − (-4))(x − (6)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve (x − (5))(x − (-8)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve (x − (2))(x − (-9)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve (x − (-1))(x − (10)) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Explain why dividing x(x − 7) = 0 by x is unsafe.
Guided self-check
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Model reasoning
Dividing by x assumes x is not zero, so it removes the valid root x = 0. The zero-product property gives x = 0 or x − 7 = 0, so the complete solution set is x = 0 or x = 7.
Write a first attempt before opening the self-check.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.