Completing the Square Calculator
Convert a quadratic ax²+bx+c to vertex form a(x−h)²+k, find the vertex (h, k), and calculate real roots if they exist.
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How to use this calculator
Completing the square rewrites the quadratic by creating a perfect square trinomial, revealing the vertex of the parabola directly.
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Enter the coefficients a, b, and c from the quadratic equation ax²+bx+c.
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The vertex form a(x−h)²+k is displayed, along with the vertex coordinates.
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If real roots exist (discriminant ≥ 0), they are shown. The axis of symmetry is x = h.
Frequently asked questions
What does "completing the square" mean?
Completing the square rewrites ax²+bx+c into the form a(x−h)²+k by adding and subtracting the square of half the x-coefficient. The result is a perfect square binomial plus a constant, which makes the vertex (h, k) obvious and enables easy graphing.
How do vertex form and standard form relate?
Both represent the same parabola. Standard form ax²+bx+c is easiest for finding y-intercept (set x=0 → y=c) and roots. Vertex form a(x−h)²+k directly reveals the vertex (h,k) and axis of symmetry (x=h). You can always expand vertex form back to standard form.
When does the quadratic have no real roots?
When the discriminant b²−4ac < 0. In vertex form: if a > 0 and k > 0, the parabola opens upward and the vertex is above the x-axis, so it never crosses. If a < 0 and k < 0, it opens downward and the vertex is below. In these cases the roots are complex (involving i = √−1).
Why is completing the square useful?
It derives the quadratic formula, enables converting circle/ellipse/parabola equations to standard form in conic sections, solves integration problems in calculus (completing the square under an integral), and is used in control theory to analyze stability of dynamical systems.
Completing the square — vertex form and roots of a quadratic
Step-by-step process
For ax²+bx+c, divide through by a: x²+(b/a)x+(c/a). Add and subtract (b/2a)²: [x+(b/2a)]²−(b/2a)²+(c/a). Multiply back by a: a[x+(b/2a)]² + c − b²/(4a). The vertex is at h = −b/(2a) and k = c − b²/(4a). This process turns the quadratic into a form where the minimum or maximum value (at the vertex) is immediately visible.
Deriving the quadratic formula
The famous quadratic formula x = (−b ± √(b²−4ac)) / (2a) is derived directly by completing the square and solving for x. Starting from a(x−h)²+k=0: (x−h)² = −k/a, so x−h = ±√(−k/a), giving x = h ± √(−k/a). Substituting h = −b/(2a) and k = c−b²/(4a) recovers the standard formula. Completing the square is therefore the foundation of the most important formula in algebra.
Applications in geometry and calculus
In geometry, completing the square is used to convert circle equations from general form (x²+y²+Dx+Ey+F=0) to standard form (x−h)²+(y−k)²=r², immediately revealing the center and radius. In calculus, it simplifies integrals of the form ∫dx/(ax²+bx+c) by transforming the denominator into a perfect square plus a constant, enabling standard integral formulas.
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