Comprehensive Notes – Algebra, Geometry & Trigonometry

Quadratic Equations

• Standard (general) form: ax^2+bx+c=0 with a\neq 0
• Terms: ax^2 = quadratic; bx = linear; c = constant
• To rewrite into standard form: collect all terms on one side, arrange as x^2,\,x,\,constant.

Solving Techniques

1. Extracting the Square Roots

• Applicable to x^2=k.
• Steps
– Isolate x^2
– Take square root of both sides → x=\pm\sqrt{k}
– Simplify radicals, rationalise denominators if needed.

2. Factoring

• Works only if the trinomial is factorable.
• Procedure

  1. Write in standard form ax^2+bx+c=0.

  2. Factor LHS.

  3. Apply Zero-Product-Property: set each factor =0.

  4. Solve for x; check in original equation.
    • Example: x^2+6=5x\;\Rightarrow\;x^2-5x+6=0\;(x-2)(x-3)=0\;\rightarrow\;x=2,\,3.

3. Completing the Square

• Re-express ax^2+bx+c=0 as (x-h)^2=k.
• Steps

  1. If a\neq1 divide both sides by a.

  2. Move constant to right.

  3. Add \big(\tfrac{b}{2a}\big)^2 to both sides.

  4. Factor perfect square; simplify right side.

  5. Extract roots, solve, verify.

4. Quadratic Formula

• Universal method, valid for any quadratic.
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
• Example: 2x^2+5x+3=0 → x=\frac{-5\pm\sqrt{1}}{4}=\left{-1,\;-\tfrac{3}{2}\right}.

Discriminant and Nature of the Roots

• Discriminant: \Delta=b^2-4ac (requires standard form).
• \Delta=0 → real, equal roots.
• \Delta>0 & perfect square → real, unequal, rational.
• \Delta>0 & not perfect square → real, unequal, irrational.
• \Delta<0 → no real roots (complex pair).
• Worked classifications:
– x^2-6x+9=0 → \Delta=0 → roots 3,3.
– x^2+11x+24=0 → \Delta=25 → rational unequal -3,-8.
– h^2+3h-16=0 → \Delta=73 → irrational unequal.
– 4k^2+6k+3=0 → \Delta=-12 → no real root.

Sum & Product of Roots

• From ax^2+bx+c=0:
– Sum x1+x2=-\dfrac{b}{a}
– Product x1x2=\dfrac{c}{a}.
• Useful to build quadratic given roots: x^2-(\text{sum})x+(\text{product})=0.

Equations Transformable to Quadratic

• Expand, collect terms, reduce to standard quadratic, then solve with preferred method.
• Ex: (x+1)^2+(x-8)^2=45 → 2x^2-14x+20=0 → factor, x=2,5.

Word-Problem Templates

• Area–perimeter pairs → use sum/product logic.
• Example pathway: width w=10 m, length l=35 m satisfy lw=350 & l+w=45.

Quadratic Inequalities

• Forms: ax^2+bx+c\,{>,<,\ge,\le}\,0.
• Solve by:

  1. Replace inequality by equality → find roots.

  2. Plot roots, divide number line into regions.

  3. Test one point per region; shade true regions.

  4. Translate to interval notation.
    • Graphical solution in 2-variables: draw parabola (dashed for strict, solid for inclusive) and shade appropriate region (inside/outside) after test point.

Quadratic Functions

• Function form: f(x)=ax^2+bx+c, a\neq0.
• Vertex (standard ⇒ vertex): f(x)=a(x-h)^2+k where h=-\tfrac{b}{2a}, k=\tfrac{4ac-b^2}{4a}.
• Domain: \mathbb R.
• Range:
– a>0: [k,\infty)
– a<0: (-\infty,k]. • Axis of symmetry: x=h. • Opening: a>0 upward, a<0 downward.
• Effect of parameters
– |a| larger → steeper (narrower); |a| smaller → wider.
– h shifts horizontally; k shifts vertically.
• To convert standard→vertex: use completing square or h,k formulas.
• Determine quadratic from graph or table by solving for a,b,c using three ordered pairs or using vertex & another point.

Variation

Direct Variation

• y=kx, k=\dfrac yx constant.
• Graph: straight line through origin with slope k.

Inverse Variation

• y=\dfrac{k}{x}, hyperbola decreasing.
• Extended form: F=\dfrac{k}{d^2} etc.

Joint Variation

• y=kxz (or variants with radicals/exponents).
• Solve k with one data set, then substitute.

Combined Variation

• Mix of direct & inverse, e.g. z=\dfrac{kx}{y}.

Exponents and Radicals

Zero & Negative Exponents

• x^0=1 (for x\neq0).
• x^{-n}=\dfrac1{x^n}.

Rational Exponents

• x^{m/n}=\sqrt[n]{x^m}=\big(\sqrt[n]{x}\big)^m.
• Laws of exponents extend to rational exponents.

Laws of Radicals

• Power rule: \big(\sqrt[n]{a}\big)^n=a.
• Product: \sqrt[n]{ab}=\sqrt[n]{a}\,\sqrt[n]{b}.
• Quotient: \sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}.
• Nested: \sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}.

Simplifying Radicals

• Factor radicand, extract perfect n^{\text{th}} powers.
• Rationalise denominator (multiply by appropriate radical to clear).
• Indices should be reduced to lowest terms.

Operations

• Add/Subtract: combine only similar radicals (same index & radicand).
• Multiply radicals of same order → multiply radicands then simplify.
• Multiply binomials with radicals → FOIL then simplify.
• Division: convert to same order, rationalise; use conjugate for binomial denominators.

Radical Equations

• Isolate radical, raise both sides to appropriate power, solve, always check for extraneous roots.
• Example: \sqrt{5x^2-16}=x → square → quadratic → x=\pm2; check → x=2 valid.

Quadrilaterals & Special Parallelograms

Classification

• Parallelogram – both pairs opposite sides parallel.
• Rectangle – parallelogram + 4 right angles.
• Rhombus – parallelogram + 4 congruent sides.
• Square – rectangle & rhombus simultaneously.
• Trapezoid – exactly one pair parallel sides; isosceles if legs congruent.
• Kite – two distinct pairs of adjacent congruent sides.

Parallelogram Properties

• Opposite sides congruent & parallel.
• Opposite angles congruent.
• Consecutive angles supplementary.
• Diagonals bisect each other.
• Each diagonal forms two congruent triangles.

Tests for Parallelogram

• Either property set guarantees figure is a parallelogram (e.g. both pairs opp. sides congruent).

Rectangle

• Diagonals congruent.
• Converse: parallelogram with congruent diagonals ⇒ rectangle.

Rhombus

• Diagonals ⟂ and each diagonal bisects opposite angles.
• Converse: parallelogram with diagonal that bisects angles ⇒ rhombus.

Square

• Inherits all rectangle & rhombus facts: right angles, sides equal, diagonals congruent & perpendicular, diagonals angle-bisecting.

Midline (Triangle Midsegment) Theorem

• Segment joining midpoints of two sides of a triangle is parallel to third side & half its length.

Trapezoids & Kites

• Isosceles trapezoid: base angles congruent, diagonals congruent.
• Median (midsegment) of trapezoid: \text{median}=\dfrac{\text{base}1+\text{base}2}{2} parallel to bases.
• Kite: diagonals ⟂; main diagonal bisects vertex angles & the other diagonal; only one pair opp. angles congruent.

Similarity & Proportions

• Proportion properties: cross-multiplication, alternation, inversion, addition/subtraction, sum of ratios.
• Similar polygons: corresponding angles congruent, corresponding sides proportional; similarity ratio \dfrac{\text{length}}{\text{length}}.

Triangle Similarity Criteria

• AA, SAS (proportional sides incl. angle), SSS (three proportional sides).
• Right-triangle special criteria:
– Acute-angle similarity (one acute angle equal).
– Leg-Leg, Hypotenuse-Leg proportionality.
• Special Right Triangles
– 45\,–\,45\,–\,90: sides 1:1:\sqrt2.
– 30\,–\,60\,–\,90: sides 1:\sqrt3:2.

Pythagorean Theorem

a^2+b^2=c^2 for right triangle; used for distance, ladders, diagonals, etc.

Trigonometry

Six Primary Ratios (for right △, reference ∠)

• \sin=\dfrac{\text{opp}}{\text{hyp}} \csc=\dfrac{\text{hyp}}{\text{opp}}
• \cos=\dfrac{\text{adj}}{\text{hyp}} \sec=\dfrac{\text{hyp}}{\text{adj}}
• \tan=\dfrac{\text{opp}}{\text{adj}} \cot=\dfrac{\text{adj}}{\text{opp}}
• Mnemonics: SOH-CAH-TOA & CHO-SHA-CAO.

Special-Angle Values

• \sin30=\tfrac12, \cos30=\tfrac{\sqrt3}{2}, \tan30=\tfrac{\sqrt3}{3}.
• \sin45=\cos45=\tfrac{\sqrt2}{2}, \tan45=1.
• \sin60=\tfrac{\sqrt3}{2}, \cos60=\tfrac12, \tan60=\sqrt3.

Angles of Elevation/Depression

• Angle measured between horizontal line and line of sight.
• Solved using \tan\theta=\dfrac{\text{opp}}{\text{adj}} or other ratios.

Law of Sines

\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}
• Cases: ASA, AAS, SSA (ambiguous).
• Can also write reciprocal form.

Law of Cosines

a^2=b^2+c^2-2bc\cos A (cyclic permutations).
• Cases: SAS, SSS; also formulas for \cos A etc.

Oblique-Triangle Problem Strategy

  1. Identify known parts vs required.

  2. Choose Law of Sines (easier) when applicable; otherwise Law of Cosines.

  3. After two angles are known, third angle =180^\circ- (A+B).

These bullet-point notes capture definitions, theorems, procedures, proofs, worked examples, and real-world applications spanning quadratic algebra, exponents & radicals, Euclidean geometry, similarity, right-triangle trigonometry, and laws for oblique triangles.