03 Quadratic inequalities · g10-03-01 · 5–10 min
Does the middle always solve it?
What changes when the leading coefficient changes sign?
Predict: For a(x−r)(x−s)≤0, keep r,s fixed and compare positive, negative and zero a.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Leading coefficient a = 1; Root r = -2; Root s = 3 | |
Case 2
Leading coefficient a = -1; Root r = -2; Root s = 3 | |
Case 3
Leading coefficient a = 0; Root r = -2; Root s = 3 | |
Revise: My claim needs the condition… because…
Exit: Solve −(x−1)(x−4)≤0.
03 Quadratic inequalities · g10-03-02 · 5–10 min
How many real zeros?
Can a quadratic be positive without crossing the axis?
Predict: Change c in ax²+bx+c; predict two, one or no real zeros before revealing Δ.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Coefficient a = 1; Coefficient b = 0; Constant c = -1 | |
Case 2
Coefficient a = 1; Coefficient b = -2; Constant c = 1 | |
Case 3
Coefficient a = 1; Coefficient b = 0; Constant c = 1 | |
Revise: My claim needs the condition… because…
Exit: How many real zeros does x²+1 have? What sign does it have?
03 Quadratic inequalities · g10-03-03 · 5–10 min
Do the endpoints belong?
What is the difference between < and ≤ at a root?
Predict: Keep the same polynomial and change only the inequality symbol.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Coefficient a = 1; Root r = 1; Root s = 4 | |
Case 2
Coefficient a = 1; Root r = 1; Root s = 4 | |
Case 3
Coefficient a = 1; Root r = 2; Root s = 2 | |
Revise: My claim needs the condition… because…
Exit: Solve (x−2)²<0 and (x−2)²≤0.
03 Quadratic inequalities · g10-03-04 · 5–10 min
A profitable value may be impossible
Which non-loss solutions are feasible in the context?
Predict: Predict the feasible set for P(x)=a(x−r)(x−s)≥0 under 0≤x≤d, then require integer x.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Coefficient a = -1; Root r = 2; Root s = 8; Capacity d = 10 | |
Case 2
Coefficient a = -1; Root r = 2; Root s = 8; Capacity d = 6 | |
Case 3
Coefficient a = -1; Root r = 3; Root s = 3; Capacity d = 6 | |
Revise: My claim needs the condition… because…
Exit: P(x)=−(x−2)(x−8), 0≤x≤6. Find feasible integer non-loss sizes.
