Completing the Square Calculator – Guide & Formulas
Complete the square for any quadratic equation. Find the vertex form, vertex coordinates, and axis of symmetry with step-by-step solutions.
Complete the square for any quadratic equation instantly with our free calculator. Enter coefficients a, b, and c to convert to vertex form, find the vertex, axis of symmetry, and roots with detailed steps.
Key Takeaway
Use the free Completing the Square Calculator to complete the square for any quadratic equation. find the vertex form, vertex coordinates, and axis of symmetry with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Completing the Square Calculator
- Enter the coefficient a (quadratic term) for ax² + bx + c.
- Enter the coefficient b (linear term).
- Enter the constant c.
- Review the vertex form, vertex coordinates, axis of symmetry, and roots with step-by-step breakdown.
The Formula
Variable Definitions
- a: Coefficient of the quadratic term (x²)
- b: Coefficient of the linear term (x)
- c: Constant term
- h: Vertex x-coordinate: -b/(2a)
- k: Vertex y-coordinate: c - b²/(4a)
Completing the Square for x² + 6x + 5
Convert x² + 6x + 5 to vertex form.
- Step 1: Identify coefficients: a = 1, b = 6, c = 5.
- Step 2: Compute h = -b/(2a) = -6/(2×1) = -3.
- Step 3: Compute k = c - b²/(4a) = 5 - 36/4 = 5 - 9 = -4.
- Step 4: Vertex form: (x + 3)² - 4. Vertex: (-3, -4).
Frequently Asked Questions
What is completing the square?
Completing the square is a method of rewriting a quadratic equation ax² + bx + c into the vertex form a(x - h)² + k, where (h, k) is the vertex of the parabola.
When is completing the square useful?
It is useful for finding the vertex of a parabola, converting to vertex form, deriving the quadratic formula, and solving quadratic equations that do not factor easily.
What is the vertex form of a quadratic?
The vertex form is a(x - h)² + k, where (h, k) is the vertex. This form makes it easy to graph the parabola and identify its minimum or maximum.
Can completing the square handle negative a values?
Yes. When a is negative, the parabola opens downward, and completing the square still works. The vertex represents the maximum point.
How do I find the axis of symmetry?
The axis of symmetry is the vertical line x = h, where h = -b/(2a). This line passes through the vertex and divides the parabola into mirror halves.
What if a is not 1?
Divide all terms by a first, then complete the square. The result is a[(x + b/(2a))² - b²/(4a²)] + c, which simplifies to a(x + b/(2a))² + c - b²/(4a).
How does this differ from the quadratic formula?
Both methods solve ax² + bx + c = 0. Completing the square is more visual and produces vertex form, while the quadratic formula directly gives the roots using x = (-b ± √(b²-4ac))/(2a).
What are the roots when completing the square?
The roots occur where a(x - h)² + k = 0, giving x = h ± √(-k/a). If -k/a is negative, the roots are complex (no real solutions).