LEARNER JOURNEY LABSFactor x² + bx + c by finding two numbers with sum b and product c.
Standalone concept packageNo prior equation-solving skill is assumed. Arithmetic operations, variables, equality, and every new algebra decision are explained on this page.
Factor x² + bx + c by finding two numbers with sum b and product c.
Construct the reasoning path, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
For x² + bx + c, choose m and n so m + n = b and mn = c. Then x² + bx + c = (x + m)(x + n).
The sum controls the middle term while the product controls the constant term.
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Factor and solve x² + 2x − 15 = 0.
The reasoning path changes as each decision is selected.
Every example identifies the goal, explains the next move, carries out the algebra, and checks or interprets the result.
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. Solve x² + 7x + 10 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
2. Solve x² − 7x + 12 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Solve x² − 5x − 6 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Solve all six. Use the specific feedback to revise a method or sign error.
1. Solve x² − 4x − 21 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
2. Solve x² + 6x − 16 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
3. Solve x² − 1x − 20 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
4. Solve x² + 7x + 6 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
5. Solve x² − 10x + 9 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
6. Solve x² + 5x − 14 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Attempt all five. At least four correct answers are required for mastery.
1. Solve x² − 10x + 24 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
2. Solve x² − 3x − 40 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
3. Solve x² + 6x − 27 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
4. Solve x² + 14x + 40 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
5. Solve x² − 13x + 22 = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Factor x² − 2x − 35 and explain how both the product and sum determine the signs.
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
The factors of 35 that differ by 2 are 5 and 7. A product of −35 requires opposite signs, and the sum must be −2, so use 5 and −7. Then (x + 5)(x − 7) = 0, giving x = −5 or x = 7.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.