Completing the square

Completing the square is a method used to rewrite a quadratic expression in a form that allows us to easily solve for \(x\).

The goal is to express a quadratic equation in the form:

\[(x + p)^2 = q\]

where \(p\) and \(q\) are constants.

Solve by completing the square

\[x^2 + 6x + 5 = 0\]

Move the constant term to the other side.

\[x^2 + 6x = -5\]

Find \(\left(\dfrac{b}{2}\right)^2\)

\[\left(\dfrac{6}{2}\right)^2 = 9\]

Add the 9 to both sides.

\[x^2 + 6x + 9 = -5 + 9\]

Rewrite the left side as a perfect square.

\[(x + 3)^2 = 4\]

Solve for \(x\).

\[x + 3 = \pm 2\]

\[x = -3 \pm 2\]

So the solutions are:

\[x = -3 + 2 = -1 \quad \text{and} \quad x = -3 - 2 = -5\]