LEARNER JOURNEY Rewrite x² + 5x = 14
Name the goalStandard form requires zero on one side.
Choose the next moveSubtract 14 from both sides.
Carry out the mathematicsx² + 5x − 14 = 0.
Check and interpretThe coefficients are a = 1, b = 5, c = −14.
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
Recognize a quadratic equation and rewrite it as ax² + bx + c = 0.
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
- ax²
- bx
- c
- = 0
- a ≠ 0
- Standard form
Read the complete visual relationship as text
- ax²
- bx
- c
- = 0
- a ≠ 0
Definition in plain language
A quadratic equation has a nonzero x² term and can be written ax² + bx + c = 0 with a ≠ 0.
Why this matters
Standard form reveals the signed coefficients used in factoring, the discriminant, and the quadratic formula.
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 every term to one side.
Order terms by descending exponent.
Insert zero coefficients for missing terms mentally.
Identify signed a, b, and c.
The same method in words
- Move every term to one side.
- Order terms by descending exponent.
- Insert zero coefficients for missing terms mentally.
- Identify signed a, b, and c.
Interactive decision model
Build the reasoning, one decision at a time
Rewrite 5 = 2x² + 3x and identify the coefficients.
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.
Identify coefficients
Name the goalUse 3x² − 8x + 2 = 0.
Choose the next moveMatch each signed term to ax² + bx + c.
Carry out the mathematicsa = 3, b = −8, c = 2.
Check and interpretThe negative sign is part of b.
Handle a missing term
Name the goalUse 4x² − 9 = 0.
Choose the next moveWrite the missing linear term as 0x.
Carry out the mathematicsa = 4, b = 0, c = −9.
Check and interpretA zero coefficient records that the term is absent.
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. Write x² + 5x = 6 in standard form.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. For 2x² − 7x + 3 = 0, what is a?
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. For 2x² − 7x + 3 = 0, what is b?
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. For 2x² − 7x + 3 = 0, what is c?
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Write a prediction before viewing the choices.
2. Which equation is quadratic?
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3. What is the degree of 5x² + 2x − 1?
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4. In standard form, what must appear on one side?
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5. Rewrite 4 = x² − 3x in standard form.
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6. For −x² + 6 = 0, what is b?
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Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. For 7x² − 9x = 0, what is c?
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Write a prediction before viewing the choices.
2. Why is x² = 9 not yet in standard form?
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3. Which value cannot be the leading coefficient a of a quadratic?
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4. What feature distinguishes a quadratic from a linear equation?
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5. Before factoring x² + 2x = 15, what standard-form equation should be used?
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Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Rewrite 7x − 2 = −3x² in standard form and identify a, b, and c.
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
Add 3x² to both sides and add 2 to both sides: 3x² + 7x + 2 = 0. Therefore a = 3, b = 7, and c = 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.