LEARNER JOURNEY Solve x² = 36
Name the goalThe square is isolated.
Choose the next moveTake both square roots.
Carry out the mathematicsx = ±6, so x = −6 or 6.
Check and interpretBoth values square to 36.
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 an isolated square by taking both the positive and negative square roots.
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.
- inverse operation
- An operation that reverses another operation, such as subtraction reversing addition.
- 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
- Isolate square
- ±√k
- Positive branch
- Negative branch
- Check both
Read the complete visual relationship as text
- Isolate square
- ±√k
- Positive branch
- Negative branch
- Check both
Definition in plain language
If (x − h)² = k with k ≥ 0, then x − h = ±√k, so x = h ± √k.
Why this matters
Squaring hides sign, so both a positive and a negative base can produce the same nonnegative square.
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.
Isolate the squared expression.
Take both square roots using ±.
Solve each resulting linear equation.
Check both candidates in the original square.
The same method in words
- Isolate the squared expression.
- Take both square roots using ±.
- Solve each resulting linear equation.
- Check both candidates in the original square.
Interactive decision model
Build the reasoning, one decision at a time
Solve (x + 4)² = 16.
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 − 2)² = 9
Name the goalThe squared base is x − 2.
Choose the next moveWrite x − 2 = ±3.
Carry out the mathematicsx = 5 or x = −1.
Check and interpretBoth are 3 units from 2.
Solve 3(x + 1)² = 75
Name the goalIsolate the square first.
Choose the next moveDivide by 3: (x + 1)² = 25, then x + 1 = ±5.
Carry out the mathematicsx = 4 or x = −6.
Check and interpretSubstitution gives 75 for both.
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 − (0))² = 9.
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 − (2))² = 16.
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 − (-3))² = 25.
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 − (5))² = 4.
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 − (-1))² = 36.
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 − (4))² = 49.
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 − (-6))² = 1.
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 − (3))² = 64.
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 − (-2))² = 81.
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))² = 9.
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 − (-5))² = 16.
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 − (1))² = 100.
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 − (8))² = 25.
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))² = 49.
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 geometrically why (x − 5)² = 49 has two solutions.
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
The equation says x is a distance 7 from 5 on the number line. Moving 7 right gives x = 12, and moving 7 left gives x = −2. Both differences, 12 − 5 and −2 − 5, square to 49.
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.