Completing the Square Calculator

Convert ax² + bx + c to vertex form a(x - h)² + k with detailed step-by-step solutions.

Enter Quadratic Coefficients

Enter values for ax² + bx + c

2x² + 8x + 3

Result

Vertex Form
2(x + 2)² - 5
a(x - h)² + k
Vertex (h, k) (-2, -5)
Axis of Symmetry x = -2
Opens Upward (a > 0)
y-intercept (0, 3)
Discriminant 40

Step-by-Step Solution

2x² + 8x + 3 = 2(x + 2)² - 5

What Is Completing the Square?

Completing the square is an algebraic technique used to rewrite a quadratic expression in standard form (ax² + bx + c) into vertex form a(x - h)² + k. This process reveals the vertex of the parabola, which is the highest or lowest point on the graph, depending on the sign of the leading coefficient.

This method is one of the most important techniques in algebra. It is used to solve quadratic equations, derive the quadratic formula, analyze the properties of parabolas, and appears in many areas of higher mathematics including calculus and linear algebra.

The Step-by-Step Process

Step 1: Factor out a

If a is not 1, factor it out from the x² and x terms (not the constant).

a(x² + (b/a)x) + c

Step 2: Find (b/2a)²

Take half of the coefficient of x inside the parentheses and square it.

(b / 2a)²

Step 3: Add and Subtract

Add and subtract this value inside the parentheses to create a perfect square trinomial.

a(x² + (b/a)x + (b/2a)² - (b/2a)²) + c

Step 4: Factor and Simplify

Factor the perfect square trinomial and distribute the subtracted term.

a(x + b/2a)² + c - b²/4a

Why Is Completing the Square Important?

Completing the square has several crucial applications in mathematics:

  • Solving quadratic equations: It can solve any quadratic, even when factoring fails.
  • Deriving the quadratic formula: The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0.
  • Graphing parabolas: Vertex form immediately reveals the vertex, axis of symmetry, and direction of opening.
  • Optimization problems: Finding maximum or minimum values in real-world applications.
  • Conic sections: Rewriting equations of circles, ellipses, and hyperbolas into standard form.

Vertex Form Properties

When a quadratic is written in vertex form a(x - h)² + k:

  • The vertex of the parabola is at the point (h, k).
  • The axis of symmetry is the vertical line x = h.
  • If a > 0, the parabola opens upward and k is the minimum value.
  • If a < 0, the parabola opens downward and k is the maximum value.
  • The value of |a| determines how wide or narrow the parabola is.

Worked Example

Let us complete the square for 3x² - 12x + 7:

  1. Factor out a = 3: 3(x² - 4x) + 7
  2. Half of -4 is -2, and (-2)² = 4
  3. Add and subtract 4 inside: 3(x² - 4x + 4 - 4) + 7
  4. Factor: 3((x - 2)² - 4) + 7
  5. Distribute: 3(x - 2)² - 12 + 7
  6. Simplify: 3(x - 2)² - 5

The vertex is at (2, -5), and the parabola opens upward since a = 3 > 0.

Common Mistakes to Avoid

  • Forgetting to factor out the leading coefficient before completing the square.
  • Not multiplying by the leading coefficient when bringing the subtracted term outside the parentheses.
  • Sign errors when the coefficient of x is negative.
  • Confusing vertex form (x - h) with (x + h) -- remember h has the opposite sign from what appears in the parentheses.