MathLesson 2 of 3

Algebra, Sets & Functions

Formulas

Quadratic Formula

x=(−b±√(b²−4ac)) / 2a

Discriminant

Δ>0→2 real rootsΔ=0→1 rootΔ<0→no real roots

Absolute Inequality

|ax+b|≤c⟺−c≤ax+b≤c

Infinite Geometric Series

S∞=a / (1−r) (|r|<1)

Logarithm Laws

ln(AB)=ln A+ln Bln(A/B)=ln A−ln Bln(Aⁿ)=n·ln Aln(eˣ)=x

Inclusion-Exclusion

|A∪B|=|A|+|B|−|A∩B|

Part 1 of 1

Quadratic & Log Equations

Quadratic formula

x=(−b±√(b²−4ac)) / 2a

Discriminant Δ = b² − 4ac tells you how many roots:

Δ>0→two distinct real rootsΔ=0→one repeated rootΔ<0→no real roots

Absolute value inequality

|ax+b|≤c⟺−c≤ax+b≤c

Log identity: ln(x) = k means x = eᵏ. If ln(x) = 3, then x = e³.

Worked example

1|2x − 5| ≤ 9.
2Apply directly: −9 ≤ 2x − 5 ≤ 9 → −4 ≤ 2x ≤ 14 → −2 ≤ x ≤ 7. log₂(x(x−2)²) = 3: x(x−2)² = 8, with x > 2.
3Try x = 4: 4·4 = 16 ≠ 8.
4Try x = 4 from quadratic factoring → x = 4. ✓

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