Paradigm A Math Consolidation Cheatsheet - Quadratic Equations & Inequalities - Nature of Roots, Surds, Polynomials

Oct 13, 2026

Math consolidation cheatsheet focused on key algebra and calculus-adjacent techniques: completing the square for equations and inequalities, dividing negative coefficients and flipping inequality signs, and solving quadratic inequality ranges. It also summarizes the nature of roots using discriminant conditions and graph behavior. Further sections cover surd rationalisation (eliminating square roots from denominators), mensuration formulas using surds, and core polynomial tools including coefficients, degree, polynomial division with missing terms, identities, remainder and factor theorems, and strategies for solving cubic factorisation and equations.

LanguageEnglish

CategoryPresentations

FormatPDF

Pages8

Reading Time3 min

AccessFree to download

Equation & Inequalities

Completing The Square

Quadratic Inequalities

Reverse Quadratic Inequalities

Nature of Roots

Proving Questions

Surds

Rationalisation of Surds

Mensuration with Surds

Polynomials

Important Terms

Solving Unknowns

Remainder & Factor Theorem

Reverse Engineer Polynomial Equation

Cubic Factorisation

Solve Cubic Equations

EQUATION & INEQUALITIES Completing The Square 􀫛􀢞􀫛 − 􀫞􀢞 + 􀫢􀫞 􀫢 = 􀫛 (􀢞􀫛 − 􀢞 + )􀫛 􀫛 = 􀫛 [(􀢞􀫛 − 􀫞􀫛 􀢞 + ( − 􀫞􀫝)􀫛 − ( − 􀫞􀫝)􀫛 + 􀫢􀫛]= 􀫛 [(􀢞 − 􀫞􀫝)􀫛 + 􀫝􀫚􀫠􀫟] = 􀫛 (􀢞 − 􀫞􀫝)􀫛 + 􀫝􀫠􀫡 Quadratic Inequalities [Divide negative, flip sign if it’s −􀢞􀫛 ] Question Type 1: Quadratic Inequalities Find the range of values of 􀝔 for which 􀝔(􀝔 − 12) ≤ 3􀝔(1 + 􀝔) Step 1: Flush everything to the left and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿 Step 2: Simplify and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿 Step 3: Solve your quadratic inequalities Question Type 2: Reverse Quadratic Inequalities Find the values of 􀝌 and 􀝍 for which 􀝔 < −3 or 􀝔 > 2 is the solution set of 􀝔 2 + 􀝍 > −􀝌􀝔 . Step 1: 􀝔 < −3 or 􀝔 > 2 Step 2: (􀝔 + 3)(􀝔 − 2) > 0 (Reverse and Form Back Original) Step 3: 􀝔 2 + 􀝔 − 6 > 0 (Expand) Step 4: 􀝔 2 − 6 > −􀝔 (Rearrange according to question) Step 5: 􀝍 = −6, 􀝌 − 1 (Compare coefficient) NATURE OF ROOTS | Determinants (Curve & Axis) | | | | --- | --- | --- | | 􀜾2 − 4􀜽􀜿 < 0
􀜾2 − 4􀜽􀜿 = 0
􀜾2 − 4􀜽􀜿 > 0
􀜾2 − 4􀜽􀜿 ≥ 0 | No Roots
1 Roots
2 Roots
1/2 Roots | No Real Roots or Imaginary Roots or
Graph is always positive (Completely above x-axis) or
Graph is always negative (Completely below x-axis) Real & Equal or
Real & Repeated Roots Real & Distinct Roots or Different Roots
Graph has real roots or Graph Intersects the x-axis | Question Type 1: Nature of Roots Find the range of 􀜽, for which the graph is Step 1: Equate Curve and Line if applicable* (=, >, <) Step 2: Flush everything to the left and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿 Step 3: Determine Nature of Roots base on the condition Step 4: Simplify and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿 Step 5: Solve your quadratic inequalities Step 6: Reject if Necessary Question Type 2: Proving Questions (Prove/Show/Explain) Show that the expression is negative for all real values of 􀝔 . Prove that the expression is always positive for all values of 􀝔 . Step 1: Flush everything to the left and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿 Step 2: Apply 􀜾2 − 4􀜽􀜿 but don’t put the sign Step 3: Simplify and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿 Step 4: Use Completing The Square/otherwise (See Condition) Step 5: For all real values of 􀝔 , is positive/negative (Proven) Example 􀜾 2 − 4􀜽􀜿 = [−(􀝇 + 2)]2 − 4(3)(2􀝇 − 9) = 􀝇 2 + 4􀝇 + 4 − 24􀝇 + 108 = 􀝇 2 − 20􀝇 + 111 = (􀝇 − 10)2 + 12 Since (􀝇 − 10)2 + 12 > 0, 􀜾 2 − 4􀜽􀜿 > 0 and line intersects the curve for all real values ofk. SURDS | Rationalisation of Surds: (We don’t like SqRoots in Denominator)
Note: Change Signs while Rationalizing | | | --- | --- | |
3 √2 3√2 ~~ ~~ × ~~ ~~ = ~~ ~~√2 √2 2

1 √3 √3 ~~ ~~ × ~~ ~~ = ~~ ~~5√3 √3 15 |
4 2 − √3 4(2 − √3) ~~ ~~ × ~~ ~~ = ~~ ~~
2 + √3 2 − √3 4 − 3 = 8 − √3 | 􀜽√􀜾 + 􀜿√􀝀 = 􀝔√􀜾 + 􀝕√􀝀 implies that 􀜽 = 􀝔 and 􀜿 = 􀝕 E.g. 3􀜽 + 􀜾 + 2􀜽√5 = −29 − 12√5 2􀜽 = −12 ⟹ 􀢇 = −􀫟 3􀜽 + 􀜾 = −29 ⟹ 3(−6) + 􀜾 = −29 ⟹ 􀢈 = −􀫚􀫚 Mensuration with Surds | Cone
Surface Area: 􀟨􀝎2 + 􀟨􀝎􀝈
Volume:13 􀟨􀝎2 ℎ | Pyramid Surface Area: Add all sides
Volume: 13 Base area x Height | | --- | --- | | Hemi-Sphere Surface Area: 2􀟨􀝎2 [open] /3􀟨􀝎2
Volume: 23 􀟨􀝎3 | Sphere Surface Area: 4􀟨􀝎2
Volume: 43 􀟨􀝎3 | | Prism Surface Area: Add all sides
Volume: Cross Section x Length | Cylinder Surface Area: 2􀟨􀝎2 + 2􀟨􀝎ℎ
Volume: 􀟨􀝎2 ℎ | POLYNOMIALS Important terms: Coefficient of a Variable: Number in front of the variable (algebra) 2 is the coefficient of 2􀝔 , 4 is the coefficient od 4􀝔2 Degree of a Polynomial: The degree of a polynomial is the highest power in the equation Adding, Subtraction & Multiplication follows algebra rules Division of Polynomials Note: When you encounter polynomials such as Divide 􀝔 3 − 9􀝔 + 10 by 􀝔 + 1 Please write it as 􀝔 3 + 0􀝔2 − 9􀝔 + 10. Fil up the missing terms with 0 Question Type 1: Solving Unknowns Method 1: Fully expand and compare coefficient Method 2: Substitution Metho

FAQs

How do you solve quadratic inequalities using this sheet’s method?

Flush all terms to the left, rearrange into the form of a quadratic using the given coefficients, then solve the quadratic inequalities. Reject cases if required.

What determines the nature of roots for a quadratic?

Use the discriminant expression: if it’s less than 0 there are no real roots; equal to 0 gives one repeated root; greater than 0 gives two real roots (distinct). Inequalities can also be interpreted via curve/axis position.

How is surd rationalisation handled here?

Rationalise by removing square roots from the denominator. Change signs appropriately while multiplying to eliminate irrational denominators, then simplify the resulting expression.

Oct 13, 2026

Upload by Taylor Morgan