Quadratic Equations and Parabolas

Quadratics and polynomialsSpecial Factoring PatternsQuadratic Equations and ParabolasPolynomial Theorems

Forms of a quadratic

Standard form

y = a x^{2} + b x + c

a ≠ 0

Vertex form

y = a (x - h)^{2} + k

vertex (h, k)

Factored (intercept) form

y = a (x - r_{1}) (x - r_{2})

x-intercepts r₁ and r₂

Completing the square

x^{2} + b x + {(\frac{b}{2})}^{2} = {(x + \frac{b}{2})}^{2}

Axis of symmetry

x = - \frac{b}{2 a} = \frac{r_{1} + r_{2}}{2}

Vertex

(- \frac{b}{2 a}, c - \frac{b^{2}}{4 a})

Quadratic formula

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Discriminant

D = b^{2} - 4 a c

D > 0, a perfect square

two different rational roots

D > 0, not a perfect square

two different irrational roots

D = 0

one repeated real root

D < 0

two complex roots (conjugates)

Sum of roots

r_{1} + r_{2} = - \frac{b}{a}

Product of roots

r_{1} r_{2} = \frac{c}{a}

Difference of roots

| r_{1} - r_{2} | = \frac{\sqrt{b^{2} - 4 a c}}{| a |}

real roots

Quadratic from its roots

x^{2} - (r_{1} + r_{2}) x + r_{1} r_{2} = 0

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