LEARNER JOURNEY Solve x² + 6x + 5 = 0
Name the goalMove 5: x² + 6x = −5.
Choose the next moveHalf 6 is 3; square it to get 9.
Carry out the mathematicsAdd 9: (x + 3)² = 4. Then x + 3 = ±2.
Check and interpretx = −1 or −5; both check.
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
Create a perfect-square trinomial and use it to solve 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.
- coefficient
- A numerical factor multiplying a variable.
- standard form
- The arrangement ax² + bx + c = 0, where a is not zero.
Look first, then name the mathematics
Build the visual meaning
- x² + 6x
- Half b = 3
- Square it = 9
- Add both sides
- Perfect square
Read the complete visual relationship as text
- x² + bx
- Half b
- Square it
- Add both sides
- Perfect square
Definition in plain language
For x² + bx, add (b/2)² to create (x + b/2)². Add the same quantity to both sides to preserve equality.
Why this matters
The method transforms a general monic quadratic into a form that can be solved by square roots.
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.
Move the constant to the other side.
Take half of b and square it.
Add that value to both sides and factor the perfect square.
Use ± square roots and solve both branches.
The same method in words
- Move the constant to the other side.
- Take half of b and square it.
- Add that value to both sides and factor the perfect square.
- Use ± square roots and solve both branches.
Interactive decision model
Build the reasoning, one decision at a time
Complete the square for x² + 10x + 9 = 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 x² − 4x − 5 = 0
Name the goalMove −5 to get x² − 4x = 5.
Choose the next moveHalf −4 is −2; square it to get 4.
Carry out the mathematicsAdd 4: (x − 2)² = 9.
Check and interpretx − 2 = ±3, so x = 5 or −1.
Solve x² + 8x + 7 = 0
Name the goalMove 7: x² + 8x = −7.
Choose the next moveHalf 8 is 4; square it to get 16.
Carry out the mathematicsAdd 16: (x + 4)² = 9.
Check and interpretx = −1 or −7, verified by substitution.
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 x² + (6)x + (5) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve x² + (4)x + (-5) = 0 by completing the square.
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 x² + (-8)x + (7) = 0 by completing the square.
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 x² + (10)x + (9) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve x² + (-2)x + (-3) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve x² + (12)x + (11) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve x² + (-6)x + (-7) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve x² + (2)x + (-8) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
6. Solve x² + (14)x + (24) = 0 by completing the square.
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 x² + (-10)x + (16) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve x² + (8)x + (-9) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve x² + (-4)x + (-12) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve x² + (16)x + (39) = 0 by completing the square.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve x² + (-12)x + (20) = 0 by completing the square.
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
Complete the square to solve x² − 12x + 20 = 0 and explain the added value.
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
Move 20: x² − 12x = −20. Half of −12 is −6, and (−6)² = 36, so add 36 to both sides: (x − 6)² = 16. Then x − 6 = ±4, giving x = 2 or x = 10.
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.