LEARNER JOURNEY Solve x² − 5x + 6 = 0
Name the goalIdentify a = 1, b = −5, c = 6.
Choose the next moveCompute D = 25 − 24 = 1.
Carry out the mathematicsx = [5 ± 1]/2.
Check and interpretThe roots are 2 and 3; 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
Use signed coefficients in the quadratic formula and simplify both 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.
- 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
- a, b, c
- −b
- Discriminant
- ± branches
- Entire numerator ÷ 2a
Read the complete visual relationship as text
- a, b, c
- −b
- Discriminant
- ± branch
- Entire numerator ÷ 2a
Definition in plain language
For ax² + bx + c = 0, x = [−b ± √(b² − 4ac)]/(2a). The formula solves every quadratic with a ≠ 0.
Why this matters
The formula packages completing-the-square reasoning into a reliable general method.
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.
Write the equation in standard form.
Record signed a, b, and c.
Evaluate the discriminant b² − 4ac.
Substitute, simplify both ± branches, and check.
The same method in words
- Write the equation in standard form.
- Record signed a, b, and c.
- Evaluate the discriminant b² − 4ac.
- Substitute, simplify both ± branches, and check.
Interactive decision model
Build the reasoning, one decision at a time
Use the formula for 2x² + x − 3 = 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 2x² − 7x + 3 = 0
Name the goalUse a = 2, b = −7, c = 3.
Choose the next moveD = 49 − 24 = 25.
Carry out the mathematicsx = [7 ± 5]/4.
Check and interpretThe roots are 1/2 and 3.
Solve 3x² + x − 2 = 0
Name the goalUse a = 3, b = 1, c = −2.
Choose the next moveD = 1 + 24 = 25.
Carry out the mathematicsx = [−1 ± 5]/6.
Check and interpretThe roots are −1 and 2/3.
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. Use the quadratic formula to solve 1x² + (-5)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.
2. Use the quadratic formula to solve 2x² + (-7)x + (3) = 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. Use the quadratic formula to solve 3x² + (1)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.
Independent practice
Six problems for repetition
Solve all six. Use the specific feedback to revise a method or sign error.
1. Use the quadratic formula to solve 2x² + (5)x + (-3) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Use the quadratic formula to solve 4x² + (-4)x + (-3) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Use the quadratic formula to solve 3x² + (-10)x + (3) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Use the quadratic formula to solve 5x² + (3)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. Use the quadratic formula to solve 2x² + (-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.
6. Use the quadratic formula to solve 6x² + (1)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. Use the quadratic formula to solve 3x² + (8)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.
2. Use the quadratic formula to solve 4x² + (4)x + (-3) = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Use the quadratic formula to solve 5x² + (-11)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.
4. Use the quadratic formula to solve 2x² + (9)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.
5. Use the quadratic formula to solve 3x² + (-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.
Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Use the quadratic formula to solve 4x² + 4x − 3 = 0 and explain the denominator.
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
Here a = 4, b = 4, c = −3. The discriminant is 16 + 48 = 64. Then x = [−4 ± 8]/8 because 2a = 8. The two roots are x = 1/2 and x = −3/2.
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.