LEARNER JOURNEY Classify x² − 5x + 6 = 0
Name the goala = 1, b = −5, c = 6.
Choose the next moveD = 25 − 24 = 1.
Carry out the mathematicsThe discriminant is positive.
Check and interpretThere are two distinct real roots.
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 b² − 4ac to predict whether a quadratic has two, one, or no real solutions.
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
- D > 0
- Two real roots
- D = 0
- One repeated root
- D < 0
- No real roots
Read the complete visual relationship as text
- D > 0
- Two real roots
- D = 0
- One repeated root
- D < 0: no real roots
Definition in plain language
The discriminant D = b² − 4ac is the quantity under the square root in the quadratic formula. D > 0 gives two real roots, D = 0 gives one repeated real root, and D < 0 gives no real roots.
Why this matters
The sign of the discriminant predicts the real-solution structure before the entire formula is simplified.
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 quadratic in standard form.
Record signed a, b, and c.
Compute D = b² − 4ac.
Classify the sign and state the number of real solutions.
The same method in words
- Write the quadratic in standard form.
- Record signed a, b, and c.
- Compute D = b² − 4ac.
- Classify the sign and state the number of real solutions.
Interactive decision model
Build the reasoning, one decision at a time
Classify 3x² + 4x + 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.
Classify x² + 6x + 9 = 0
Name the goala = 1, b = 6, c = 9.
Choose the next moveD = 36 − 36 = 0.
Carry out the mathematicsThe discriminant is zero.
Check and interpretThere is one repeated real root, x = −3.
Classify x² + 2x + 5 = 0
Name the goala = 1, b = 2, c = 5.
Choose the next moveD = 4 − 20 = −16.
Carry out the mathematicsThe discriminant is negative.
Check and interpretThere are no real roots because a real square root of −16 does not exist.
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 discriminant to classify the solutions of 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 discriminant to classify the solutions of 1x² + (2)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.
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 discriminant to classify the solutions of 1x² + (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.
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 discriminant to classify the solutions of 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.
2. Use the discriminant to classify the solutions of 3x² + (4)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.
3. Use the discriminant to classify the solutions of 4x² + (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.
4. Use the discriminant to classify the solutions of 2x² + (1)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.
5. Use the discriminant to classify the solutions of 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.
6. Use the discriminant to classify the solutions of 3x² + (6)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.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. Use the discriminant to classify the solutions of 2x² + (4)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. Use the discriminant to classify the solutions of 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.
3. Use the discriminant to classify the solutions of 1x² + (-8)x + (16) = 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 discriminant to classify the solutions of 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.
5. Use the discriminant to classify the solutions of 4x² + (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.
Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Compare the solution counts of x² − 4x + 4 = 0 and x² − 4x + 5 = 0 without fully solving either.
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
For x² − 4x + 4, D = 16 − 16 = 0, so there is one repeated real root. For x² − 4x + 5, D = 16 − 20 = −4, so there are no real roots.
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.