LEARNER JOURNEY Solve x² + 6x = 0
Name the goalBoth terms share x.
Choose the next moveFactor x to get x(x + 6) = 0.
Carry out the mathematicsx = 0 or x + 6 = 0, so x = 0 or −6.
Check and interpretBoth roots satisfy the original equation.
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
Factor the greatest common factor before solving a quadratic.
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
- Shared factor
- Factor out
- 3x(2x + 3)
- Zero product
- Roots
Read the complete visual relationship as text
- Shared factor
- Factor out GCF
- Product = 0
- x = 0
- Other root
Definition in plain language
A greatest common factor divides every term. Factoring it out rewrites a sum as a product without changing its value.
Why this matters
The GCF often exposes a zero root and makes the remaining factor linear or easier to factor.
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.
Set the polynomial equal to zero.
Find the greatest factor shared by every term.
Rewrite as a product.
Set each factor equal to zero and solve.
The same method in words
- Set the polynomial equal to zero.
- Find the greatest factor shared by every term.
- Rewrite as a product.
- Set each factor equal to zero and solve.
Interactive decision model
Build the reasoning, one decision at a time
Solve 4x² + 20x = 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 3x² − 12x = 0
Name the goalThe terms share 3x.
Choose the next moveFactor 3x(x − 4) = 0.
Carry out the mathematics3x = 0 gives x = 0; x − 4 = 0 gives x = 4.
Check and interpretThe numerical factor 3 does not create an additional root.
Solve −2x² − 8x = 0
Name the goalA negative GCF can make the inner leading term positive.
Choose the next moveFactor −2x(x + 4) = 0.
Carry out the mathematicsx = 0 or x = −4.
Check and interpretSubstitution verifies both roots.
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 2x² + (6)x = 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 3x² + (-12)x = 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 5x² + (10)x = 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 -2x² + (8)x = 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 4x² + (-20)x = 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 6x² + (18)x = 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 -3x² + (-15)x = 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 7x² + (14)x = 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 8x² + (-24)x = 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 -5x² + (25)x = 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 9x² + (27)x = 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 10x² + (-40)x = 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 -4x² + (12)x = 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 12x² + (36)x = 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
Solve 6x² − 18x = 0 without dividing by x.
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
Factor 6x(x − 3) = 0. Then 6x = 0 gives x = 0, and x − 3 = 0 gives x = 3. Factoring preserves the zero root that division by x could remove.
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.