Describe the process of completing the square.

Completing the square is a method used to rewrite a quadratic expression in a standard form.

To complete the square, we start with a quadratic expression in the form of ax^2 + bx + c. We then take half of the coefficient of x, square it, and add it to both sides of the equation. This creates a perfect square trinomial on the left-hand side of the equation.

For example, let's say we have the quadratic expression x^2 + 6x + 5. We take half of the coefficient of x, which is 3, and square it, which gives us 9. We then add 9 to both sides of the equation:

x^2 + 6x + 9 = -5 + 9

On the left-hand side, we now have a perfect square trinomial, which can be factored as (x + 3)^2. On the right-hand side, we simplify to get:

(x + 3)^2 = 4

Finally, we take the square root of both sides of the equation, remembering to include both the positive and negative square roots:

x + 3 = ±2

We then solve for x by subtracting 3 from both sides:

x = -3 ± 2

So the solutions to the original quadratic equation are x = -1 and x = -5.

Completing the square is a useful method for solving quadratic equations, as it allows us to rewrite the equation in a standard form that is easy to factor or solve using the quadratic formula. It is also used in other areas of mathematics, such as calculus and differential equations.