Completing the Square

Step-by-step solution

1. 1

   The quadratic

   Rewrite the quadratic expression in vertex form a(x − h)² + k, reading off a = 1, b = 6 and c = 5.

   $$x^{2} + 6x + 5$$
2. 2

   Halve the x-coefficient and square it

   Half of 6 is 3; squaring it gives 9. This is the number that completes the square inside the bracket.

   $$\left(\frac{6}{2}\right)^{2} = \left(3\right)^{2} = 9$$
3. 3

   Add inside the bracket, compensate outside

   Adding 9 inside the bracket adds 9 to the whole expression (the bracket is multiplied by 1), so subtract 9 outside to keep everything unchanged.

   $$1\left(x^{2} + 6x + 9\right) + 5 - 9$$
4. 4

   Write the bracket as a perfect square

   The bracket is a perfect square: x² + 6x + 9 = (x + 3)², which is (x − h)² with h = -3.

   $$x^{2} + 6x + 5 = \left(x + 3\right)^{2} - 4$$
5. 5

   Vertex form, vertex and axis of symmetry

   With h = -3 and k = -4 the vertex is (-3, -4) and the axis of symmetry is the vertical line x = -3.

   $$x^{2} + 6x + 5 = \left(x + 3\right)^{2} - 4, \qquad \text{vertex} = \left(-3,\ -4\right)$$