Vertex form is a(x-h)²+k. The vertex of the parabola is (h,k).

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Completing the Square Calculator

Convert ax² + bx + c to vertex form a(x - h)² + k

a =

b =

c =

Completing Square Formula

ax²+bx+c = a(x + b/2a)² + c - b²/4a

Completing the square rewrites a quadratic from standard form into vertex form. The squared part shows the horizontal shift of the parabola, while the remaining constant gives the vertical shift. This makes it easy to identify the vertex, axis of symmetry, and minimum or maximum value. The same method also explains where the quadratic formula comes from, because solving a completed-square equation only requires isolating the squared term and taking square roots.

⚠Note: a must not be 0. Completing the square applies to quadratic expressions.

Why Use Vertex Form?

Graphing

a(x-h)²+k gives the vertex and opening direction directly.

Optimization

The vertex gives the maximum or minimum value of a quadratic.

Solving

After rewriting, isolate the squared term to solve equations.

Formula Derivation

Completing the square is the standard derivation of the quadratic formula.

💡 Example: x²+6x+5 = (x+3)²-4, so the vertex is (-3,-4).

Applications of Completing the Square

Vertex Form Graphing Parabolas Optimization Quadratic Formula

Frequently Asked Questions

What is a completing the square calculator?▼

A completing the square calculator rewrites ax²+bx+c as a(x-h)²+k and shows the vertex form steps.

What is vertex form?▼

Vertex form is a(x-h)²+k, where (h,k) is the vertex.

Why complete the square?▼

It makes graphing, optimization, and equation solving easier.

What is completing the square?▼

Completing the square rewrites ax²+bx+c as a(x-h)²+k, making the vertex and graph behavior easier to see.

Why divide by a first?▼

When a is not 1, factor a from the x² and x terms before completing the square inside the parentheses.

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