Completing the Square Calculator: Vertex Form a(x − h)² + k with Steps
Enter a, b and c. The calculator rewrites the expression in vertex form, shows each step of completing the square and solves it for x.
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Vertex form
2(x + 3)² − 8
- Vertex (h, k)
- (−3, −8)
- Roots (set to 0)
- −1, −5
a > 0, so the smallest value is −8, reached at x = −3.
Show working
- 2x² + 12x + 10
- 2[x² + 6x + 5]
- (b/2a)² = (3)² = 9
- x² + 6x + 9 = (x + 3)²
- 2[(x + 3)² − 9 + 5]
- = 2(x + 3)² − 8
- 2(x + 3)² − 8 = 0 → (x + 3)² = 4
- x = −3 ± √4 = −1, −5
How to use
- Enter a, b and c of ax² + bx + c.
- Read the vertex form and the vertex.
- Open “Show working” to see how (b/2a)² is added and taken away.
Good to know
Steps: take a out of the x terms, add and subtract (b/2a)² inside the bracket, and write x² + (b/a)x + (b/2a)² as (x + b/2a)². The result is a(x + b/2a)² + (c − b²/4a).
Vertex form shows the turning point straight away: in a(x − h)² + k the vertex is (h, k). If a > 0, k is the minimum value of the expression; if a < 0 it is the maximum.
Setting the vertex form to zero gives (x − h)² = −k/a, which leads directly to the quadratic formula. That is how the formula is proved in NCERT Class 10.
Questions
What does completing the square mean?
Rewriting x² + bx as a perfect square minus a number: x² + bx = (x + b/2)² − (b/2)².
What is the vertex form of 2x² + 12x + 10?
2(x + 3)² − 8. The vertex is (−3, −8), and the minimum value is −8.
Can I use it when a is not 1?
Yes. The calculator first takes a out as a common factor, as you would by hand.
What if the roots are complex?
If (x − h)² equals a negative number, there are no real roots. The calculator shows the complex roots instead.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.