AMC Formulas

Quadratic Formula

-b±sqrt(b²-4ac)/2a

Discriminant

b²-4ac

Discriminant equals 0

Single repeated real root

Discriminant is less than 0

No real roots

Discriminant is greater than 0

Two real roots

Sum of quadratic roots

-b/a

Product of quadratic roots

c/a

When do you use Vieta’s?

When you want to find the value of an expression containing the roots, not the roots themselves

Sum of reciprocals of roots equals

Sum of the roots divided by product of roots

When is a special case that you shouldn’t use Vieta’s?

When the discriminant is zero, because of the repeated root

x^(-a)

1/x^(a)

x^(a)*x^(b)

x^(a+b)

x^(a)/x^(b)

x^(a-b)

(x^(a))^(b)

x^(a*b)

x²-y²

(x-y)(x+y)

(x+y)² (fully expanded)

x²+2xy+y²

(x+y)² (in relation to x-y)

(x-y)²+4xy

(x-y)² (fully expanded)

x²-2xy+y²

(x-y)² (in relation to x+y)

(x+y)²-4xy

(x+y)²+(x-y)²

2(x²+y²)

(x+y)²-(x-y)²

4xy

(x+y+z)²

x²+y²+z²+2(xy+yz+xz)

xy+kx+jy+jk (Simon’s Favorite Factoring Trick)

(x+j)(y+k)

x^4+4y^4

(x²-2xy+2y²)(x²+2xy+2y²)

(x+y)³ (with x+y term)

x³+3xy(x+y)+y³

(x-y)³ (with x-y term)

x³-3xy(x-y)-y³

(x+y)³ (fully expanded)

x³+3x²y+3xy²+y³

(x-y)³ (fully expanded)

x³-3x²y+3xy²-y³

x³-y³

(x-y)(x²+xy+y²)

x³+y³

(x+y)(x²-xy+y²)

x³+y³+z³

(x+y+z)(x²+y²+z²-xy-xz-yz)+3xyz