Algebra Reference
Every algebra formula you need in one organized table — the quadratic formula, exponent and logarithm rules, factoring and special products, radicals, lines and distance, and complex numbers — with one-tap links to our interactive equation solver and calculators so you can work your own problem right away.
Quadratic
- x (for ax²+bx+c=0)=(−b ± √(b²−4ac)) / 2a
- Discriminant D=b² − 4ac (D>0: 2 real, D=0: 1, D<0: complex)
- Sum of roots=−b/a
- Product of roots=c/a
- Vertex x=−b/2a
Exponent Rules
- aᵐ·aⁿ=aᵐ⁺ⁿ
- aᵐ/aⁿ=aᵐ⁻ⁿ
- (aᵐ)ⁿ=aᵐⁿ
- (a·b)ⁿ=aⁿ·bⁿ
- (a/b)ⁿ=aⁿ/bⁿ
- a⁰=1 (a ≠ 0)
- a⁻ⁿ=1/aⁿ
- a^(1/n)=ⁿ√a
- a^(m/n)=ⁿ√(aᵐ)
Logarithm Properties
- logₐ(xy)=logₐx + logₐy
- logₐ(x/y)=logₐx − logₐy
- logₐ(xⁿ)=n·logₐx
- logₐa=1
- logₐ1=0
- logₐx=ln x / ln a (change of base)
- a^(logₐx)=x
Factoring & Special Products
- a² − b²=(a+b)(a−b)
- a² + 2ab + b²=(a+b)²
- a² − 2ab + b²=(a−b)²
- a³ + b³=(a+b)(a²−ab+b²)
- a³ − b³=(a−b)(a²+ab+b²)
- (a+b)³=a³+3a²b+3ab²+b³
- (a−b)³=a³−3a²b+3ab²−b³
Radicals
- √(ab)=√a·√b
- √(a/b)=√a/√b (b > 0)
- ⁿ√(aᵐ)=a^(m/n)
- √a·√a=a (a ≥ 0)
- a/√b=a√b/b (rationalize)
Lines & Distance
- slope m=(y₂−y₁)/(x₂−x₁)
- slope-intercept=y = mx + b
- point-slope=y − y₁ = m(x − x₁)
- distance d=√((x₂−x₁)² + (y₂−y₁)²)
- midpoint=((x₁+x₂)/2, (y₁+y₂)/2)
Complex Numbers
- i²=−1
- (a+bi) + (c+di)=(a+c) + (b+d)i
- (a+bi)(c+di)=(ac−bd) + (ad+bc)i
- |a+bi|=√(a²+b²)
- conjugate: (a+bi)(a−bi)=a²+b²
These are the standard algebra identities. Every one holds for all admissible values, so you can check it by plugging in numbers — for example a²−b² = (a+b)(a−b) at a=5, b=3 gives 16 = 8·2. ✓
Quadratic
- x (for ax²+bx+c=0) = (−b ± √(b²−4ac)) / 2a
- Discriminant D = b² − 4ac (D>0: 2 real, D=0: 1, D<0: complex)
- Sum of roots = −b/a
- Product of roots = c/a
- Vertex x = −b/2a
Exponent Rules
- aᵐ·aⁿ = aᵐ⁺ⁿ
- aᵐ/aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- (a·b)ⁿ = aⁿ·bⁿ
- (a/b)ⁿ = aⁿ/bⁿ
- a⁰ = 1 (a ≠ 0)
- a⁻ⁿ = 1/aⁿ
- a^(1/n) = ⁿ√a
- a^(m/n) = ⁿ√(aᵐ)
Logarithm Properties
- logₐ(xy) = logₐx + logₐy
- logₐ(x/y) = logₐx − logₐy
- logₐ(xⁿ) = n·logₐx
- logₐa = 1
- logₐ1 = 0
- logₐx = ln x / ln a (change of base)
- a^(logₐx) = x
Factoring & Special Products
- a² − b² = (a+b)(a−b)
- a² + 2ab + b² = (a+b)²
- a² − 2ab + b² = (a−b)²
- a³ + b³ = (a+b)(a²−ab+b²)
- a³ − b³ = (a−b)(a²+ab+b²)
- (a+b)³ = a³+3a²b+3ab²+b³
- (a−b)³ = a³−3a²b+3ab²−b³
Radicals
- √(ab) = √a·√b
- √(a/b) = √a/√b (b > 0)
- ⁿ√(aᵐ) = a^(m/n)
- √a·√a = a (a ≥ 0)
- a/√b = a√b/b (rationalize)
Lines & Distance
- slope m = (y₂−y₁)/(x₂−x₁)
- slope-intercept = y = mx + b
- point-slope = y − y₁ = m(x − x₁)
- distance d = √((x₂−x₁)² + (y₂−y₁)²)
- midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Complex Numbers
- i² = −1
- (a+bi) + (c+di) = (a+c) + (b+d)i
- (a+bi)(c+di) = (ac−bd) + (ad+bc)i
- |a+bi| = √(a²+b²)
- conjugate: (a+bi)(a−bi) = a²+b²
About the Algebra Reference
This is the algebra cheat sheet students reach for most: the quadratic formula, the exponent and logarithm rules, factoring and special-product identities, radicals, the equations of lines with distance and midpoint, and the basics of complex numbers — organized exactly the way they are taught. Browse the cards by topic: the quadratic formula x = (−b ± √(b²−4ac))/2a together with the discriminant D = b²−4ac and the sum and product of roots; the seven exponent laws (aᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ and more); the logarithm properties (logₐ(xy) = logₐx + logₐy, the power rule and change of base); the factoring identities including the difference of squares a²−b² = (a+b)(a−b) and the sum and difference of cubes; radical rules and rationalizing; slope, slope-intercept, point-slope, distance and midpoint; and complex-number arithmetic with the modulus and conjugate. Every formula is written in clean Unicode notation — no images — so it copies cleanly into your notes and is easy for assistive technology and AI to read. What sets this page apart from a static PDF is that it is wired into UnitConv's interactive calculators: when you find the formula you need, tap "Solve a quadratic," "Exponent calculator" or "Logarithm calculator" to open our step-by-step tools and run your actual problem. Use it as a homework reference, an exam revision list, or a quick lookup while you are working through an algebra set.
How to use this reference
- 1 Find the topic you need in the grouped cards — quadratic, exponents, logarithms, factoring, radicals, lines or complex numbers.
- 2 Copy the formula in clean Unicode notation straight into your work, or check your own result against it.
- 3 Tap "Solve a quadratic," "Exponent calculator" or "Logarithm calculator" to open the interactive tool and run your own numbers.
How these algebra formulas fit together
Most of algebra is built from a handful of identities that you can always verify by substitution. The quadratic formula x = (−b ± √(b²−4ac))/2a comes from completing the square on ax²+bx+c = 0, and its discriminant D = b²−4ac tells you the nature of the roots (two real, one repeated, or a complex conjugate pair) before you even solve. The exponent laws and the logarithm properties are two sides of the same coin — because a logarithm is the inverse of an exponential, logₐ(xy) = logₐx + logₐy is exactly the statement that aᵐ·aⁿ = aᵐ⁺ⁿ. The special-product and factoring identities are the same equation read in both directions: expanding (a+b)(a−b) gives a²−b², and factoring a²−b² gives (a+b)(a−b). The line formulas all follow from the slope m = (y₂−y₁)/(x₂−x₁), and the distance formula is just the Pythagorean theorem applied to the coordinate differences. Complex numbers extend all of this by defining i² = −1, which is what makes every quadratic solvable. Because each of these is an identity, you can confirm it numerically: pick any a and b and check, for instance, that a³−b³ really equals (a−b)(a²+ab+b²).
Frequently asked questions
What is the quadratic formula and when do I use it?
The quadratic formula solves any equation of the form ax²+bx+c = 0: x = (−b ± √(b²−4ac))/2a. Use it whenever a quadratic will not factor nicely. The quantity under the root, the discriminant D = b²−4ac, tells you what to expect: D > 0 gives two distinct real solutions, D = 0 gives one repeated real solution, and D < 0 gives a pair of complex-conjugate solutions. For example, x²−5x+6 = 0 gives x = (5 ± √1)/2 = {3, 2}.
What are the exponent rules I need to memorize?
The core laws are the product rule aᵐ·aⁿ = aᵐ⁺ⁿ, the quotient rule aᵐ/aⁿ = aᵐ⁻ⁿ, the power rule (aᵐ)ⁿ = aᵐⁿ, and the distribution rules (a·b)ⁿ = aⁿ·bⁿ and (a/b)ⁿ = aⁿ/bⁿ. Add the special cases a⁰ = 1 (for a ≠ 0), the negative exponent a⁻ⁿ = 1/aⁿ, and the fractional exponents a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(aᵐ). Together these let you simplify any expression involving powers and roots.
How are logarithm properties related to exponent rules?
A logarithm is the inverse of an exponential, so every log property mirrors an exponent law. Because aᵐ·aⁿ = aᵐ⁺ⁿ, taking logs turns multiplication into addition: logₐ(xy) = logₐx + logₐy. Likewise logₐ(x/y) = logₐx − logₐy mirrors the quotient rule, and logₐ(xⁿ) = n·logₐx mirrors the power rule. The change-of-base formula logₐx = ln x / ln a lets you compute any logarithm on a calculator that only has ln or log₁₀.
What is the difference of squares and why is it so useful?
The difference of squares is the identity a²−b² = (a+b)(a−b). It is one of the most useful factoring patterns because it turns a subtraction of two perfect squares into a product you can simplify or cancel — for example x²−9 = (x+3)(x−3). It also underlies rationalizing denominators with the conjugate: multiplying by (a−b)/(a−b) clears a radical because (a+√b)(a−√b) = a²−b. The related sum and difference of cubes, a³±b³ = (a±b)(a²∓ab+b²), extend the same idea to third powers.
Related tools
Turn any of these formulas into a worked solution on UnitConv: solve a quadratic or any equation step by step with the equation solver, evaluate powers with the exponent calculator, work with logarithms and change of base using the logarithm calculator, and tidy up very large or very small numbers with the scientific notation converter.