About

Completing the square converts a standard quadratic ax² + bx + c into vertex form a(x − h)² + k. The procedure is mechanical but error-prone: a sign mistake when halving the linear coefficient or forgetting to multiply the correction term by a will silently produce a wrong vertex. This calculator performs exact rational arithmetic with full GCD-reduced fractions and outputs every intermediate algebraic step. It assumes real coefficients with a ≠ 0.

Results include the vertex (h, k), axis of symmetry x = h, direction of opening, and exact roots (real or complex). Fractional and radical values are simplified. Note: this tool works with rational inputs only. Irrational coefficients like π must be approximated as decimals before entry.

Formulas

The completing-the-square procedure transforms the general quadratic into vertex form by algebraic manipulation of the linear and constant terms.

ax² + bx + c = a(x − h)² + k

where the vertex coordinates are:

h = −b2a

k = c − b²4a

The discriminant Δ = b² − 4ac determines root nature. When Δ > 0, two distinct real roots exist. When Δ = 0, one repeated real root. When Δ < 0, two complex conjugate roots.

x = −b ± √Δ2a

Where a = coefficient of x², b = coefficient of x, c = constant term, h = x-coordinate of vertex, k = y-coordinate of vertex, Δ = discriminant.

Reference Data

Standard Form	Vertex Form	Vertex (h, k)	Axis of Symmetry	Opens
x² + 6x + 5	(x + 3)² − 4	(−3, −4)	x = −3	Upward
2x² − 8x + 3	2(x − 2)² − 5	(2, −5)	x = 2	Upward
−1x² + 4x − 7	−1(x − 2)² − 3	(2, −3)	x = 2	Downward
x² − 10x + 25	(x − 5)²	(5, 0)	x = 5	Upward
3x² + 12x + 7	3(x + 2)² − 5	(−2, −5)	x = −2	Upward
−2x² + 6x	−2(x − 1.5)² + 4.5	(1.5, 4.5)	x = 1.5	Downward
x² + 2x + 5	(x + 1)² + 4	(−1, 4)	x = −1	Upward
5x² − 20x + 15	5(x − 2)² − 5	(2, −5)	x = 2	Upward
4x² + 4x + 1	4(x + 0.5)²	(−0.5, 0)	x = −0.5	Upward
x² − 1	(x)² − 1	(0, −1)	x = 0	Upward
−3x² + 18x − 24	−3(x − 3)² + 3	(3, 3)	x = 3	Downward
0.5x² + 3x + 1	0.5(x + 3)² − 3.5	(−3, −3.5)	x = −3	Upward

Frequently Asked Questions

When a = 0, the expression is linear (bx + c), not quadratic. Completing the square requires a nonzero leading coefficient because the procedure divides by a. The calculator rejects a = 0 with an error message.

The calculator uses exact rational arithmetic internally. When b/a is not an integer, the half-coefficient b2a is displayed as a reduced fraction using GCD simplification. For example, a = 2, b = 3 yields h = −3/4 displayed as a proper fraction in the steps.

This is the core algebraic identity: adding (b/2a)² creates a perfect square trinomial, but to preserve equality you must subtract the same quantity. When a ≠ 1, the subtracted term is multiplied by a before combining with c, which is a common source of manual errors.

Yes. When Δ < 0, the roots are expressed as h ± √−Δ/2a ⋅ i, where i is the imaginary unit. The calculator displays both roots in a + bi form with simplified radicals.

The calculator factors the radicand by extracting perfect square factors. For example, √72 becomes 6√2 because 72 = 36 × 2. This uses iterative trial division to find the largest perfect square divisor.

The sign of a determines concavity. When a > 0, the parabola opens upward and k is the global minimum. When a < 0, it opens downward and k is the global maximum. The vertex form itself is algebraically identical regardless of sign.