Quadratic Equations

Quadratic Equations

Standard form:

ax^2 + bx + c = 0

Roots:

x = [-b +/- sqrt(b^2 - 4ac)] / 2a
Sum of roots = -b/a
Product of roots = c/a
Discriminant D = b^2 - 4ac

Discriminant meaning:

D > 0: two real roots
D = 0: equal real roots
D < 0: no real roots

Factorization shortcut:

For x^2 + bx + c:
Find p and q such that p + q = b and pq = c

Signs of roots:

Both positive: sum positive, product positive
Both negative: sum negative, product positive
Opposite signs: product negative

Useful identities:

(a+b)^2 = a^2 + 2ab + b^2
(a-b)^2 = a^2 - 2ab + b^2
a^2 - b^2 = (a-b)(a+b)
(a+b)^3 = a^3 + b^3 + 3ab(a+b)
(a-b)^3 = a^3 - b^3 - 3ab(a-b)
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
a^3 - b^3 = (a-b)(a^2 + ab + b^2)