Quadratics cheat sheet

Every quadratic question reduces to one of three tasks: find the roots, find the vertex, or describe the graph. One line of algebra covers each.

Practise quadratics

Roots

$a x^{2} + b x + c = 0$ Standard form
$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ Quadratic formula
$x_{1} + x_{2} = - \frac{b}{a}, x_{1} x_{2} = \frac{c}{a}$ Vieta's formulas

Discriminant

$Δ = b^{2} - 4 a c$
$Δ > 0$ Two distinct real roots
$Δ = 0$ One repeated real root
$Δ < 0$ Two complex conjugate roots

Graph

$x = - \frac{b}{2 a}$ Axis of symmetry
$(- \frac{b}{2 a}, c - \frac{b ^{2}}{4 a})$ Vertex
$y = a (x - h)^{2} + k$ Vertex form
$y = a (x - r_{1}) (x - r_{2})$ Factored form

Completing the square

$x^{2} + b x = (x + \frac{b}{2})^{2} - \frac{b ^{2}}{4}$
$x^{2} + 6 x + 2 = (x + 3)^{2} - 7$