Completing the Square

Definition

Completing the square is a method used to solve quadratic equations and to express quadratic expressions in a perfect square form. This process involves rewriting a quadratic equation of the form:

ax^2 + bx + c = 0

in such a way that one side of the equation becomes a perfect square trinomial. The general steps for completing the square are:

1. Make sure the coefficient of x^2 is 1 (if it's not, divide the entire equation by a, the coefficient of x^2.

2. Move the constant term to the other side of the equation.

3. Add and subtract a specific number to make the left side a perfect square trinomial. This number is found by taking half of the coefficient of x, squaring it, and adding it to both sides.

4. Factor the trinomial as a perfect square.

5. Solve for x by taking the square root of both sides of the equation.

For example, to complete the square for x^2 + 6x - 7 = 0:

1. Move the constant: x^2 + 6x = 7.

2. Take half of 6 (which is 3), square it to get 9, and add 9 to both sides:  

   x^2 + 6x + 9 = 7 + 9, which simplifies to x^2 + 6x + 9 = 16.

3. Factor the left side: (x + 3)^2 = 16.

4. Solve for x by taking the square root of both sides:  

   x + 3 = +4 or -4, so x = 1 or x = -7.

This process allows you to easily solve quadratic equations and is foundational in understanding the properties of quadratic functions.

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