Charts & references · 3 min read
Algebra Formulas Cheat Sheet: Identities, Exponents, Quadratics
Special products
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- a² − b² = (a + b)(a − b)
- (a + b)³ = a³ + 3a²b + 3ab² + b³
Check numerically: with a = 4, b = 3 the left side is 49 and the right side is 49.
Exponents and logs
- aᵐ · aⁿ = aᵐ⁺ⁿ · aᵐ/aⁿ = aᵐ⁻ⁿ · (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 · a⁻ⁿ = 1/aⁿ · a^(1/n) = n-th root of a
- log(xy) = log x + log y · log(x/y) = log x − log y · log(xʸ) = y log x
See exponents explained and logarithms explained.
Quadratics
x = (−b ± √(b² − 4ac)) / 2a, with discriminant D = b² − 4ac.
x² − 5x + 6 = 0
- Factored form: (x − 3)(x − 2)
- Discriminant D = 1
- Vertex: (2.5, −0.25)
Step-by-step working 6 steps
- 1Identify a, b and c
x² − 5x + 6 = 0 a = 1, b = −5, c = 6
- 2Calculate the discriminant
D = b² − 4ac = (−5)² − 4(1)(6) = 1
D is positive: two different real roots.
- 3Apply the quadratic formula
x = (−b ± √D) / 2a = (5 ± √1) / 2 = (5 ± 1) / 2
- 4Solve for both roots
x₁ = 3 x₂ = 2
- 5Check by substitution
f(3) = 0 f(2) = 0
Both should be 0 (allowing for tiny rounding).
- 6Vertex of the parabola
x = −b / 2a = 2.5, y = −0.25 → vertex (2.5, −0.25)
The parabola opens upward, so the vertex is the minimum point.
Lines
- Slope: m = (y₂ − y₁)/(x₂ − x₁)
- Slope-intercept form: y = mx + b
- Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)
- Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Example: the distance from (1, 2) to (4, 6) is √(3² + 4²) = 5. Plot lines in the graphing calculator.
Frequently asked questions
What is the difference of squares?
a² − b² factors as (a + b)(a − b).
How do I remember the quadratic formula?
Write it out each time you solve one; after a dozen problems it sticks.
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