ScientificCalcul

Algebra

Completing the Square Explained

Completing the square adds the term needed to turn x^2 + bx into a perfect-square binomial. The method works because the same amount is added to both sides of an equation.

Reviewed

The missing term

Expanding (x + p)^2 gives x^2 + 2px + p^2. To match x^2 + bx, choose p = b/2, so the required constant is (b/2)^2.

This pattern applies directly when the coefficient of x^2 is 1. If it is not 1, divide the equation by that coefficient or factor it from the quadratic terms first.

Formulax^2 + bx + (b/2)^2 = (x + b/2)^2

Solving an equation

Move the constant term away from the x terms, add the completing term to both sides, factor the left side, and take both square roots. The plus-or-minus step is essential because positive and negative numbers can have the same square.

  1. Isolate x^2 + bx.
  2. Add (b/2)^2 to both sides.
  3. Rewrite the left side as a squared binomial.
  4. Take both square roots and isolate x.

From standard form to vertex form

Writing y = ax^2 + bx + c as y = a(x - h)^2 + k exposes the vertex (h, k) and the axis x = h. This makes graph transformations clearer than standard form.

The coefficient a is unchanged, so the direction and vertical scale of the parabola remain visible.

Formulaax^2 + bx + c = a(x + b/(2a))^2 + c - b^2/(4a)

Where this idea is used

  • Deriving the quadratic formula.
  • Finding a parabola's vertex and axis of symmetry.
  • Integrating Gaussian-type expressions and analyzing distance formulas.

Worked examples

Solve x^2 + 6x - 7 = 0

Complete the square on x^2 + 6x.

  1. x^2 + 6x = 7
  2. Add (6/2)^2 = 9 to both sides
  3. (x + 3)^2 = 16
  4. x + 3 = +/-4

Result: x = 1 or x = -7

Write x^2 - 8x + 3 in vertex form

Add and subtract 16 inside the expression.

  1. x^2 - 8x + 16 - 16 + 3
  2. (x - 4)^2 - 13

Result: y = (x - 4)^2 - 13, with vertex (4, -13)

Common mistakes

  • Using b^2/2 instead of (b/2)^2.
  • Adding the completing term to only one side of an equation.
  • Taking a square root without including both positive and negative branches.

Questions about completing the square

Can every quadratic be completed to a square?

Yes, provided the quadratic coefficient is nonzero. The result may include fractions.

When is completing the square better than factoring?

It is useful when a quadratic does not factor cleanly or when vertex form is the goal.

Why divide by a first?

The basic perfect-square pattern assumes the coefficient of x^2 is 1.

Sources and further reading