formula.expert

The quadratic family

quadratic formuladiscriminantVieta's formulasvertex formulacompleting the squaresum and product of roots

The two roots, the discriminant that predicts them, the vertex of the parabola and the sum-and-product shortcuts — all from ax² + bx + c.

Discriminant of a Quadratic

Δ=b2−4ac\Delta = b^2 - 4ac

The quantity b² − 4ac under the square root, which reveals how many real roots a quadratic has before you solve it.

Quadratic Formula (Positive Root)

x=−b+b2−4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

The larger of the two solutions of ax² + bx + c = 0, taking the plus branch of the square root in the quadratic formula.

Quadratic Formula (Negative Root)

x=−b−b2−4ac2ax = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

The smaller of the two solutions of ax² + bx + c = 0, taking the minus branch of the square root in the quadratic formula.

Sum of the Roots of a Quadratic

S=−baS = -\frac{b}{a}

Vieta's relation giving the sum of a quadratic's two roots directly from its coefficients, without solving the equation.

Product of the Roots of a Quadratic

P=caP = \frac{c}{a}

Vieta's relation giving the product of a quadratic's two roots directly from its coefficients, without solving the equation.

Vertex x-Coordinate of a Parabola

h=−b2ah = -\frac{b}{2a}

The x-coordinate of a parabola's vertex, sitting midway between the two roots and marking the axis of symmetry.

Vertex y-Coordinate of a Parabola

k=c−b24ak = c - \frac{b^2}{4a}

The y-coordinate of a parabola's vertex — its minimum value when a is positive, and its maximum when a is negative.

Completing the Square: the Constant Needed

k=(b2)2k = \left( \frac{b}{2} \right)^{2}

The constant you add to x² + bx to turn it into a perfect square trinomial, the pivotal step in completing the square.

How they fit together

All of these fall out of completing the square on ax² + bx + c, which is what the quadratic formula is: the general case, done once, so nobody has to do it again. The discriminant b² − 4ac is the part under the root, and it alone decides the character of the answer — positive gives two real roots, zero gives one repeated root where the parabola just kisses the axis, negative gives a complex pair. The vertex sits at x = −b/2a, exactly halfway between the roots, because the whole picture is symmetric about it.

Check the discriminant before you solve; it costs one subtraction and tells you whether a real answer exists at all. Use the sum and product of roots — Vieta's relations, −b/a and c/a — when the question asks about the roots collectively rather than individually, which turns several exam problems into one line. Use the vertex when you want a maximum or a minimum: projectile apex, maximum revenue, least material. The two recurring errors are forgetting to write the equation in standard form with everything on one side before reading off a, b and c, and losing the sign of b in −b/2a when b is already negative.