Algebra

Quadratic Equations formulas

At a glance

x = (−b ± √D) / 2a,   D = b² − 4ac
D < 0 none · D = 0 equal · D > 0: rational if D is a perfect square, else p ± √q
α + β = −b/a   αβ = c/a   (α − β)² = (α + β)² − 4αβ
roots α, β  →  x² − (α + β)x + αβ = 0
cubic ax³ + bx² + cx + d:  Σα = −b/a   Σαβ = c/a   αβγ = −d/a
a > 0: negative between the roots, positive outside
vertex x = −b/2a, extreme value −D/4a

Roots by formula, and the picture behind them

Roots are where the parabola meets the x-axis, symmetric about x = −b/2a

The ± gives two roots placed symmetrically about the vertex x = −b/2a.

x = (−b ± √(b² − 4ac)) / 2a

The discriminant: nature of the roots

Nature of the roots of ax² + bx + c = 0, with a, b, c rational

D = b² − 4ac	Roots	Parabola	Example
D < 0	No real roots (a complex pair)	Never meets the x-axis	x² + 2x + 5: D = −16
D = 0	Two equal real roots, x = −b/2a	Touches the x-axis at its vertex	x² − 6x + 9: D = 0, root 3
D > 0, a perfect square	Two distinct rational roots	Crosses the x-axis twice	x² − 5x + 6: D = 1, roots 2, 3
D > 0, not a perfect square	Two distinct irrational roots, a pair p ± √q	Crosses the x-axis twice	x² − 6x + 7: D = 8, roots 3 ± √2

The rational/irrational split needs rational a, b, c. With rational coefficients, irrational roots always come in conjugate pairs: if 3 + √2 is a root, so is 3 − √2.

Shortcut: If a and c have opposite signs, −4ac is positive, so D > 0 automatically. Real roots, no working needed.

Sum and product of roots

With s = α + β and p = αβ:

α + β = −b/a        αβ = c/a
α² + β² = s² − 2p
(α − β)² = s² − 4p
α³ + β³ = s³ − 3ps
1/α + 1/β = s/p

Forming an equation from its roots

From ax² + bx + c = 0 with roots α, β:

roots 1/α, 1/β    →  cx² + bx + a = 0
roots −α, −β      →  ax² − bx + c = 0
roots kα, kβ      →  ax² + kbx + k²c = 0
roots α + k, β + k →  put (x − k) in place of x

Cubics and quartics: the same pattern

Roots α, β, γ (and δ) of the equation on the left.

ax³ + bx² + cx + d = 0:   α + β + γ = −b/a    αβ + βγ + γα = c/a    αβγ = −d/a
ax⁴ + bx³ + cx² + dx + e = 0:   Σα = −b/a   Σαβ = c/a   Σαβγ = −d/a   αβγδ = e/a
α² + β² + γ² = (Σα)² − 2Σαβ        1/α + 1/β + 1/γ = Σαβ / αβγ

Shortcut: If the roots are in AP, call them a − d, a, a + d: their sum is 3a, so the middle root falls out of −b/a at once. In GP, call them a/r, a, ar: their product is a³, so the middle root comes from −d/a.

Sign of a quadratic: the wavy curve

The wavy curve: the sign flips at each simple root

Maximum and minimum at the vertex

vertex at x = −b/2a
extreme value = c − b²/4a = −D/4a
complete the square: x² − 8x + 21 = (x − 4)² + 5 ≥ 5

Shortcut: For a product of two brackets, the vertex is the average of the roots. No differentiation, no completing the square.

Formulas stick when you use them

The Quadratic Equations lesson has worked problems and quick checks for every formula here. Free to start.

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