Completing the Square
Rewrite any quadratic ax^2+bx+c into vertex form a(x-h)^2+k by completing the square. Get the vertex (h,k), axis of symmetry, and step-by-step transformation.
About This Calculator
Completing the square is a fundamental algebraic technique that rewrites any quadratic expression ax^2 + bx + c into vertex form a(x - h)^2 + k. This transformation reveals the vertex (h, k) -- the maximum or minimum point of the parabola -- and makes it easy to solve quadratic equations without relying solely on the quadratic formula.
The key insight is that a perfect square trinomial x^2 + 2px + p^2 can be factored as (x + p)^2. By adding and subtracting (b/(2a))^2, we create a perfect square within the expression. The vertex h = -b/(2a) represents the x-coordinate of the parabola's turning point, and k = c - b^2/(4a) is the y-coordinate.
This technique is particularly useful for graphing quadratic functions, determining maximum and minimum values, solving optimization problems, and deriving the quadratic formula itself. It is an essential skill in algebra, calculus, and many applied mathematics fields.
Frequently Asked Questions
How do you complete the square?
For ax^2 + bx + c = 0, factor out a, then add and subtract (b/(2a))^2 inside the parentheses. This creates a perfect square trinomial that can be written as a(x - h)^2 + k where h = -b/(2a) and k = c - b^2/(4a).
What is vertex form?
Vertex form is a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This makes it easy to identify the turning point and axis of symmetry x = h.
Why is completing the square useful?
Completing the square lets you rewrite any quadratic into vertex form, solve quadratic equations without the formula, derive the quadratic formula itself, and find maximum/minimum values of quadratic functions.
What is the axis of symmetry?
The axis of symmetry is the vertical line x = h where h = -b/(2a). It divides the parabola into two mirror-image halves and passes through the vertex.
Can you complete the square when a is not 1?
Yes, factor out the coefficient a from the x^2 and x terms first: a(x^2 + (b/a)x) + c. Then complete the square inside the parentheses and adjust the constant term outside.
How do you solve equations by completing the square?
Rewrite ax^2 + bx + c = 0 as a(x - h)^2 + k = 0, then isolate the squared term: (x - h)^2 = -k/a. Take the square root of both sides to get x = h +/- sqrt(-k/a).
What does completing the square reveal about a graph?
Vertex form immediately shows the vertex (h,k) and the axis of symmetry. The sign of a tells whether the parabola opens upward (a > 0) or downward (a < 0).
Is this calculator free to use?
Yes, all calculators on Calculy are completely free to use with no limits or registration required.