Algebra/Completing the Square
Derivation[edit]
The purpose of "completing the square" is to either factor a prime quadratic equation or to more easily graph a parabola. The procedure to follow is as follows for a quadratic equation :
1. Divide everything by a, so that the number in front of is a perfect square (1):
2. Now we want to focus on the term in front of the x. Add the quantity to both sides:
3. Now notice that on the right, the first three terms factor into a perfect square:
Multiply this back out to convince yourself that this works.
4. Therefore the completed square form of the quadratic is:
- or, multiplying through by a,
Explanation of Derivation[edit]
1. Divide everything by a, so that the number in front of is a perfect square (1):
Think of this as expressing your final result in terms of 1 square x. If your initial equation is
2. Now we want to focus on the term in front of the x. Add the quantity to both sides:
3. Now notice that on the right, the first three terms factor into a perfect square:
Multiply this back out to convince yourself that this works.
4. Therefore the completed square form of the quadratic is:
- or, multiplying through by a,
Example[edit]
The best way to learn to complete a square is through an example. Suppose you want to solve the following equation for x.
2x^{2} + 24x + 23 = 0 | Does not factor easily, so we complete the square. |
x^{2} + 12x + 23/2 = 0 | Make coefficient of x^{2} a 1, by dividing all terms by 2. |
x^{2} + 12x = - 23/2 | Add – 23/2 to both sides. |
x^{2} + 12x + 36 = - 23/2 + 36 | Take half of 12 (coefficient of x), and square it. Add to both sides. |
(x + 6)^{2} = 49/2 | Factor. Now we can take square roots to easily solve this form of the equation. |
√(x + 6)^{2} = √49/√2 | Take the square root. |
x + 6 = 7/√2 | Simplify. |
x = -6 + (7√2)/2 | Rationalize the denominator. |