How to Use
Type the values of a, b, and c from your quadratic ax² + bx + c. 'a' must not be zero.
Use "Vertex Form" to convert to a(x+h)² + k, or "Solve by Completing" to find the roots step by step.
Click Calculate to see the vertex form, vertex coordinates, axis of symmetry, and complete step-by-step working.
Frequently Asked Questions
Completing the square rewrites a quadratic ax² + bx + c into a(x + h)² + k, called vertex form. It works by adding and subtracting the value (b/2a)² to create a perfect square trinomial. It reveals the parabola's vertex and is used to solve quadratics, derive the quadratic formula, and sketch graphs.
Vertex form is a(x − h)² + k, where (h, k) is the vertex of the parabola. The vertex is the minimum point if a > 0 (opens up) or the maximum point if a < 0 (opens down). The axis of symmetry is the vertical line x = h.
For ax² + bx + c:
- Factor out a from x terms: a(x² + (b/a)x) + c
- Compute (b/2a)²
- Add and subtract it inside: a(x² + (b/a)x + (b/2a)² − (b/2a)²) + c
- Rewrite: a(x + b/2a)² + (c − b²/4a)
Yes. Once in vertex form a(x + h)² + k = 0, isolate the squared term: (x + h)² = −k/a. Take square roots: x + h = ±√(−k/a). Solve: x = −h ± √(−k/a). This is mathematically equivalent to the quadratic formula. Use the "Solve by Completing" tab above.
The axis of symmetry is the vertical line x = −b/(2a). It passes through the vertex and divides the parabola into two mirror-image halves. If you reflect any point on the parabola across this line, you get another point on the parabola.
Completing the square is used to: (1) find the vertex of a parabola for maximum/minimum problems, (2) derive the quadratic formula, (3) convert circles and ellipses from general to standard form, (4) solve equations where factoring is not obvious. It appears in JEE, CBSE Class 10–11, and many competitive exams.