Quadratic Functions (Copy)
SIMULTANEOUS EQUATIONS, QUADRATICS, INEQUALITIES & INTERSECTIONS — FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Linear Equation | y = mx + c | Mixing gradient and intercept |
| Gradient | m = (y₂ − y₁)/(x₂ − x₁) | Reversing subtraction incorrectly |
| Quadratic Equation | ax² + bx + c = 0 | Forgetting a ≠ 0 |
| Quadratic Formula | x = (−b ± √(b² − 4ac))/2a | Missing brackets in denominator |
| Discriminant | Δ = b² − 4ac | Forgetting square on b |
| Nature of Roots | Δ > 0 → 2 real roots | Mixing conditions |
| Nature of Roots | Δ = 0 → repeated root | Forgetting tangent meaning |
| Nature of Roots | Δ < 0 → no real roots | Trying to force real answers |
| Vertex Formula | x = −b/2a | Missing negative sign |
| Axis of Symmetry | x = −b/2a | Writing y instead of x |
| y-intercept | Put x = 0 | Putting y = 0 instead |
| x-intercepts | Put y = 0 | Mixing intercept types |
| Upward Parabola | a > 0 | Saying maximum instead of minimum |
| Downward Parabola | a < 0 | Saying minimum instead of maximum |
| Simultaneous Equations | Solve both together | Solving separately |
| Substitution Method | Replace one variable fully | Partial substitution |
| Elimination Method | Remove one variable | Wrong multiplication |
| Intersection Points | Same x and y satisfy both equations | Giving only x-values |
| Tangent Condition | Δ = 0 | Thinking two intersections exist |
| Line and Curve | Substitute line into curve | Substituting wrong equation |
| Circle Equation | x² + y² = r² | Forgetting squares |
| Quadratic Inequality | Solve corresponding equation first | Solving directly without roots |
| Interval Testing | Test values between roots | Using roots themselves incorrectly |
| Greater Than Zero | Graph above x-axis | Choosing below x-axis |
| Less Than Zero | Graph below x-axis | Choosing above x-axis |
| Repeated Root | (x − a)² = 0 | Giving two different roots |
| Factorisation | (x + a)(x + b) | Wrong sign combinations |
| Completing Square | (x + b/2)² − (b/2)² | Forgetting subtraction term |
| Complex Roots | x = a ± bi | Saying “no roots exist” |
| Graph Intersections | Solve simultaneously | Drawing only one graph |
| Maximum/Minimum Value | Use x = −b/2a first | Using root instead |
| Substitution into Vertex | Put h into equation for k | Forgetting second step |
| Inequality Symbols | >, <, ≥, ≤ | Missing equal sign cases |
| Solution Notation | x ∈ intervals | Writing equations instead |
| Union Notation | ∪ means “or” | Mixing with intersection |
| Infinity Notation | (−∞, ∞) | Using square brackets with ∞ |
| Perfect Square | (x − a)² | Expanding incorrectly |
| Quadratic Graph | Parabola | Drawing straight line |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK QUADRATIC FEATURES TABLE
| Feature | Formula |
|---|---|
| General Form | y = ax² + bx + c |
| Vertex x-coordinate | x = −b/2a |
| Discriminant | Δ = b² − 4ac |
| Quadratic Formula | x = (−b ± √Δ)/2a |
| Axis of Symmetry | x = −b/2a |
| y-intercept | (0, c) |
| Upward Opening | a > 0 |
| Downward Opening | a < 0 |
QUICK INEQUALITY RULES
| Situation | Solution Type |
|---|---|
| Quadratic > 0 | Above x-axis |
| Quadratic < 0 | Below x-axis |
| Δ > 0 | Two roots |
| Δ = 0 | One repeated root |
| Δ < 0 | No real roots |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK SIMULTANEOUS EQUATION STEPS
| Step | Action |
|---|---|
| 1 | Rearrange equation |
| 2 | Substitute/eliminate |
| 3 | Form quadratic |
| 4 | Solve for x |
| 5 | Find y-values |
| 6 | Write coordinates |
QUICK INTERSECTION RULES
| Situation | Meaning |
|---|---|
| Two intersections | Δ > 0 |
| Tangent | Δ = 0 |
| No intersection | Δ < 0 |
| Line and curve | Solve simultaneously |
| Same coordinates | Common solution |
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Tangent question | Check if Δ = 0 |
| Vertex question | Use x = −b/2a immediately |
| Inequality question | Sketch sign pattern |
| Simultaneous equation | Substitute line into curve |
| Upward parabola | Minimum point |
| Downward parabola | Maximum point |
| Repeated root | Graph touches axis once |
| No real roots | Graph never touches x-axis |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Missing brackets in quadratic formula | Wrong answers throughout |
| Wrong discriminant sign | Wrong nature of roots |
| Forgetting to substitute for y | Incomplete coordinates |
| Mixing x and y intercepts | Incorrect graph features |
| Wrong interval selection | Entire inequality wrong |
| Arithmetic sign errors | Final answer incorrect |
| Forgetting ± sign | Missing root |
| Not simplifying roots | Accuracy marks lost |
| Ignoring repeated root | Wrong number of solutions |
| Incorrect expansion | Entire quadratic incorrect |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.