Completing the square is a useful method in algebra for rewriting a quadratic equation in a standard form. To complete the square for any quadratic expression of the form ax² + bx + c, you want to transform it into a perfect square trinomial.
The first step is to focus on the b coefficient and follow these steps:
- Start with the quadratic expression: ax² + bx + c.
- If a is not 1, factor it out of the first two terms: a(x² + (b/a)x) + c.
- To complete the square, take half of the b/a, square that value, and add it inside the parentheses. The number to add is ((b/2a)²).
- Because you added a value inside the parentheses, you need to adjust the equation by subtracting the same value multiplied by a outside the parentheses. Thus, you actually add (b/2)²/a to both sides of the equation.
Consequently, the number you should add to both sides of the equation when completing the square is:
((b/2)² / a)
By completing the square, you can rewrite the original equation in a form that makes it easier to solve or analyze the properties of the quadratic function it represents.
For example, for the quadratic equation x² + 6x + 5, we would take:
Half of 6 is 3. Squaring it gives you 9.So, you add 9 to both sides:
x² + 6x + 9 = 5 + 9This transforms it to:
(x + 3)² = 14