Completing the Square Calculator — Show Steps

How Completing the Square Works

Completing the square is an algebraic technique that transforms a quadratic expression from standard form (ax^2 + bx + c) into vertex form (a(x - h)^2 + k), revealing the vertex coordinates (h, k) of the corresponding parabola. This method is one of the foundational techniques in algebra, appearing in curricula aligned with the Common Core State Standards for Mathematics (standards HSA-SSE.B.3 and HSA-REI.B.4). It is tested on the SAT, ACT, GRE, and AP Calculus exams, and is essential for graphing parabolas, solving quadratic equations, and working with conic sections.

The technique works by manipulating the expression so that the x terms form a perfect square trinomial. Once in vertex form, you can immediately read the vertex (h, k), the axis of symmetry (x = h), and whether the parabola opens upward (a > 0) or downward (a < 0). The vertex form is also the most efficient way to find the minimum or maximum value of a quadratic function, which has applications in physics (projectile motion), economics (profit maximization), and engineering (optimization). The quadratic formula is itself derived by completing the square on the general quadratic equation.

The Completing the Square Formula

For a quadratic expression ax^2 + bx + c, the vertex form is obtained using these relationships:

h = -b / (2a) (x-coordinate of the vertex)

k = c - b^2 / (4a) (y-coordinate of the vertex)

Vertex Form: a(x - h)^2 + k

For example, given x^2 - 6x + 5 (where a = 1, b = -6, c = 5): h = -(-6) / (2 x 1) = 3, and k = 5 - (-6)^2 / (4 x 1) = 5 - 9 = -4. The vertex form is (x - 3)^2 - 4, with vertex at (3, -4). The axis of symmetry is x = 3, and the minimum value of the function is -4. Setting (x - 3)^2 - 4 = 0 gives x = 3 +/- 2, so x = 1 or x = 5 (the roots).

Key Terms You Should Know

Standard Form: A quadratic expression written as ax^2 + bx + c, where a, b, and c are constants and a is not equal to 0. This is the most common form encountered in textbooks and the starting point for completing the square.

Vertex Form: The equivalent expression a(x - h)^2 + k, where (h, k) is the vertex. This form is optimal for graphing because it immediately reveals the vertex, axis of symmetry, and direction of opening.

Discriminant: The value b^2 - 4ac, which determines the nature of the roots. A positive discriminant means two distinct real roots, zero means one repeated root (the vertex touches the x-axis), and negative means two complex conjugate roots (the parabola does not cross the x-axis).

Perfect Square Trinomial: An expression of the form x^2 + 2dx + d^2, which factors as (x + d)^2. Completing the square works by creating a perfect square trinomial from the x terms.

Axis of Symmetry: The vertical line x = h that divides the parabola into two mirror-image halves. Every parabola has exactly one axis of symmetry passing through its vertex.

Completing the Square vs Other Methods

Different methods for solving quadratics have distinct advantages. The following comparison, consistent with how these methods are taught in standard algebra textbooks such as those from the Mathematical Association of America, helps determine which approach to use.

Practical Worked Examples

Example 1 — Simple case (a = 1): Complete the square for x^2 + 8x + 12. Step 1: h = -8/(2x1) = -4. Step 2: k = 12 - 64/4 = 12 - 16 = -4. Step 3: Vertex form is (x + 4)^2 - 4. The vertex is (-4, -4). The roots are x = -4 +/- 2 = -2 and -6. Check: (-2)^2 + 8(-2) + 12 = 4 - 16 + 12 = 0. Correct.

Example 2 — Leading coefficient not 1: Complete the square for 2x^2 - 12x + 7. Factor out 2: 2(x^2 - 6x) + 7. Half of -6 is -3, squared is 9. Add and subtract 9: 2(x^2 - 6x + 9 - 9) + 7 = 2(x - 3)^2 - 18 + 7 = 2(x - 3)^2 - 11. The vertex is (3, -11). The minimum value is -11. The graphing calculator can visualize this parabola.

Example 3 — Negative leading coefficient: Complete the square for -3x^2 + 24x - 10. Factor out -3: -3(x^2 - 8x) - 10. Half of -8 is -4, squared is 16. We get -3(x^2 - 8x + 16 - 16) - 10 = -3(x - 4)^2 + 48 - 10 = -3(x - 4)^2 + 38. The vertex is (4, 38). Since a = -3 < 0, this is a maximum. The maximum value of the function is 38, occurring at x = 4.

Tips for Completing the Square Successfully

• Always factor out the leading coefficient first: If a is not equal to 1, factor it from the x^2 and x terms before adding and subtracting. Forgetting this step is the most common error students make.
• Remember to distribute when subtracting: When you add (b/2a)^2 inside parentheses that have a coefficient, the subtracted value is multiplied by that coefficient. For 3(x^2 + 4x + 4 - 4), the -4 becomes -12 when distributed, not -4.
• Use the shortcut formulas for quick answers: h = -b/(2a) and k = c - b^2/(4a) give the vertex directly without going through all the steps. This is faster on timed exams like the SAT and ACT.
• Check your answer by expanding: Expand the vertex form back to standard form and verify it matches the original expression. This catches sign errors and arithmetic mistakes.
• Practice with the equation solver: Compare your manual work against the calculator to build confidence and identify where errors occur in your process.

Applications Beyond Algebra

Completing the square extends far beyond solving simple quadratic equations. In physics, it determines the maximum height and range of projectile motion (h(t) = -16t^2 + v0t + h0 completed into vertex form gives maximum height and time of flight). In economics, it finds the quantity that maximizes profit or minimizes cost in quadratic revenue and cost models. In analytic geometry, completing the square on x and y terms separately converts the general second-degree equation Ax^2 + Bx + Cy^2 + Dy + E = 0 into the standard forms of circles ((x-h)^2 + (y-k)^2 = r^2), ellipses, and hyperbolas. In calculus, completing the square is a prerequisite technique for integration of expressions like 1/(x^2 + bx + c) using the arctangent formula.