Completing the Square Calculator

Completing the square is a method for rewriting any quadratic expression from standard form (ax squared plus bx plus c) into vertex form (a(x minus h) squared plus k). The vertex form immediately reveals the vertex of the parabola at coordinates (h, k): h is the axis of symmetry and k is the minimum value of the function if a is positive or the maximum if a is negative. This technique underpins the derivation of the quadratic formula, the analysis of conic sections, and techniques in calculus such as integration of rational functions. It is also the most systematic way to solve a quadratic equation when factoring is not obvious: once you have the vertex form you simply isolate x and take the square root of both sides. This calculator accepts the three coefficients a, b and c of a quadratic in the form ax squared plus bx plus c, rewrites it in vertex form, identifies the vertex and the discriminant, and solves for the roots if they exist. It also shows each intermediate step. The defaults use a=1, b=-6, c=5, the quadratic x squared minus 6x plus 5, which factors as (x minus 1)(x minus 5) and has roots x=1 and x=5.

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Coefficient a

Coefficient b

Coefficient c

(x - 3)² - 4

vertex form

Vertex (h, k)(3, -4)

Discriminant16

Root 1x = 1

Root 2x = 5

How it works

For ax squared plus bx plus c, the vertex h equals minus b divided by 2a. Vertex k equals c minus b squared divided by 4a. These give the vertex form a(x minus h) squared plus k. The discriminant is b squared minus 4ac. If the discriminant is positive there are two real roots given by the quadratic formula: x = (minus b plus or minus the square root of the discriminant) divided by 2a. If zero, one repeated root. If negative, no real roots. Each step is built up from the intermediate values and displayed below the result card.

Worked example

Quadratic: x squared minus 6x plus 5 (a=1, b=-6, c=5). h = -(-6)/(2 times 1) = 6/2 = 3. k = 5 - (-6) squared/(4 times 1) = 5 - 36/4 = 5 - 9 = -4. Vertex form: (x - 3) squared - 4. Discriminant: (-6) squared - 4(1)(5) = 36 - 20 = 16. Roots: (6 plus or minus 4)/2, giving x = 5 and x = 1. These match the defaults pre-filled above.

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Frequently asked questions

What does completing the square do?

Completing the square rewrites a quadratic expression in standard form (ax squared plus bx plus c) into vertex form a(x minus h) squared plus k. This reveals the vertex of the parabola at (h, k) and makes it straightforward to read off the axis of symmetry, maximum or minimum value, and roots.

How do you complete the square step by step?

For x squared plus bx plus c: take half of b (call it p), then add and subtract p squared. The expression becomes (x plus p) squared minus p squared plus c. For example, x squared minus 6x plus 5 becomes (x minus 3) squared minus 9 plus 5, which is (x minus 3) squared minus 4.

Can you complete the square when a is not 1?

Yes. Factor out a from the first two terms first: ax squared plus bx plus c equals a(x squared plus (b/a)x) plus c. Then complete the square inside the bracket and adjust the constant term accordingly. The vertex h is always minus b divided by 2a, and k is c minus b squared divided by 4a.