Completing the Square Calculator: Vertex Form, Vertex, Roots and Discriminant of a Quadratic

How this calculator works

Enter a, b and c from ax^2 + bx + c.

The calculator completes the square to give the vertex form.

It also gives the vertex, the discriminant and the roots.

Formulas used

• ax^2 + bx + c = a(x - h)^2 + k
• h = -\frac{b}{2a},\ k = c - \frac{b^2}{4a}
• x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Worked example

With the default numbers in the calculator above:

1. Vertex form: 2 (x - 3)^2 - 8
2. Vertex x (h): 3
3. Vertex y (k): -8
4. Discriminant: 64
5. Root 1: 5
6. Root 2: 1
7. y-intercept: 10

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: a (the number in front of x squared) 1; b (the number in front of x) 2; c (the constant) 5.

1. Vertex form: 1 (x + 1)^2 + 4
2. Vertex x (h): -1
3. Vertex y (k): 4
4. Discriminant: -16
5. Root 1 (complex): -1 + 2i
6. Root 2 (complex): -1 - 2i
7. y-intercept: 5

Common questions

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• Synthetic Division Calculator: Divide a Polynomial by (x - c) with Quotient and Remainder

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any algebra text (completing the square, the quadratic formula).

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