Quadratic Formula Sheet — Equation, Discriminant & Forms A complete reference for the quadratic equation ax² + bx + c = 0. Includes the quadratic formula, discriminant rules, vertex and standard forms, factoring patterns, completing the square, and Vieta's formulas for the sum and product of roots.
The Quadratic Formula Standard Form
a x 2 + b x + c = 0 ( a ≠ 0 ) a x^2 + b x + c = 0 \quad(a \ne 0) a x 2 + b x + c = 0 ( a = 0 ) x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c Discriminant
Δ = b 2 − 4 a c \Delta = b^2 - 4ac Δ = b 2 − 4 a c Discriminant: Nature of Roots Two Complex (Conjugate) Roots
Perfect Square (Rational Roots)
Δ = k 2 for some integer k \Delta = k^2 \text{ for some integer } k Δ = k 2 for some integer k Forms of a Quadratic Standard Form
y = a x 2 + b x + c y = a x^2 + b x + c y = a x 2 + b x + c Vertex Form
y = a ( x − h ) 2 + k y = a(x - h)^2 + k y = a ( x − h ) 2 + k Factored Form
y = a ( x − r 1 ) ( x − r 2 ) y = a(x - r_1)(x - r_2) y = a ( x − r 1 ) ( x − r 2 ) Vertex Coordinates
( h , k ) = ( − b 2 a , c − b 2 4 a ) (h, k) = \left(-\tfrac{b}{2a},\ c - \tfrac{b^2}{4a}\right) ( h , k ) = ( − 2 a b , c − 4 a b 2 ) Axis of Symmetry
x = − b 2 a x = -\frac{b}{2a} x = − 2 a b Completing the Square Step 1: Divide by a
x 2 + b a x = − c a x^2 + \tfrac{b}{a} x = -\tfrac{c}{a} x 2 + a b x = − a c Step 2: Add (b/2a)²
x 2 + b a x + ( b 2 a ) 2 = − c a + ( b 2 a ) 2 x^2 + \tfrac{b}{a} x + \left(\tfrac{b}{2a}\right)^2 = -\tfrac{c}{a} + \left(\tfrac{b}{2a}\right)^2 x 2 + a b x + ( 2 a b ) 2 = − a c + ( 2 a b ) 2 Step 3: Factor
( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 \left(x + \tfrac{b}{2a}\right)^2 = \tfrac{b^2 - 4ac}{4a^2} ( x + 2 a b ) 2 = 4 a 2 b 2 − 4 a c Result
x = − b 2 a ± b 2 − 4 a c 4 a 2 x = -\tfrac{b}{2a} \pm \sqrt{\tfrac{b^2 - 4ac}{4a^2}} x = − 2 a b ± 4 a 2 b 2 − 4 a c Vieta's Formulas (Sum & Product of Roots) Sum of Roots
r 1 + r 2 = − b a r_1 + r_2 = -\frac{b}{a} r 1 + r 2 = − a b Product of Roots
r 1 ⋅ r 2 = c a r_1 \cdot r_2 = \frac{c}{a} r 1 ⋅ r 2 = a c Reconstruct from Roots
x 2 − ( r 1 + r 2 ) x + r 1 r 2 = 0 x^2 - (r_1 + r_2) x + r_1 r_2 = 0 x 2 − ( r 1 + r 2 ) x + r 1 r 2 = 0 Common Factoring Patterns Difference of Squares
a 2 − b 2 = ( a + b ) ( a − b ) a^2 - b^2 = (a + b)(a - b) a 2 − b 2 = ( a + b ) ( a − b ) Perfect Square Trinomial (+)
a 2 + 2 a b + b 2 = ( a + b ) 2 a^2 + 2 a b + b^2 = (a + b)^2 a 2 + 2 ab + b 2 = ( a + b ) 2 Perfect Square Trinomial (−)
a 2 − 2 a b + b 2 = ( a − b ) 2 a^2 - 2 a b + b^2 = (a - b)^2 a 2 − 2 ab + b 2 = ( a − b ) 2 Sum of Cubes
a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) a^3 + b^3 = (a + b)(a^2 - a b + b^2) a 3 + b 3 = ( a + b ) ( a 2 − ab + b 2 ) Difference of Cubes
a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a^3 - b^3 = (a - b)(a^2 + a b + b^2) a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) Special Cases & Applications Projectile Motion
h ( t ) = − 1 2 g t 2 + v 0 t + h 0 h(t) = -\tfrac{1}{2} g t^2 + v_0 t + h_0 h ( t ) = − 2 1 g t 2 + v 0 t + h 0 Time to Hit Ground
0 = − 1 2 g t 2 + v 0 t + h 0 ⟹ t = v 0 + v 0 2 + 2 g h 0 g 0 = -\tfrac{1}{2} g t^2 + v_0 t + h_0 \implies t = \tfrac{v_0 + \sqrt{v_0^2 + 2 g h_0}}{g} 0 = − 2 1 g t 2 + v 0 t + h 0 ⟹ t = g v 0 + v 0 2 + 2 g h 0 Maximum Height (Projectile)
h_\max = h_0 + \tfrac{v_0^2}{2 g}
Pure Quadratic (b = 0)
a x 2 + c = 0 ⟹ x = ± − c a a x^2 + c = 0 \implies x = \pm \sqrt{-\tfrac{c}{a}} a x 2 + c = 0 ⟹ x = ± − a c 