Algebra/Completing the Square

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Algebra
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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  y = ax^2+bx+c :

1. Divide everything by a, so that the number in front of  x^2 is a perfect square (1):

 \frac{y}{a} = x^2 + \frac{b}{a}x + \frac{c}{a}

2. Now we want to focus on the term in front of the x. Add the quantity  \left(\frac{b}{2a}\right)^2 to both sides:

 \frac{y}{a} + \left(\frac{b}{2a}\right)^2 = x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 + \frac{c}{a}

3. Now notice that on the right, the first three terms factor into a perfect square:

 x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 = \left(x + \frac{b}{2a}\right)^2

Multiply this back out to convince yourself that this works.

4. Therefore the completed square form of the quadratic is:

 \frac{y}{a} + \left(\frac{b}{2a}\right)^2 = \left(x + \frac{b}{2a}\right)^2 + \frac{c}{a} or, multiplying through by a,

 y = a\left(x+\frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

Explanation of Derivation[edit]

Completing The Square

1. Divide everything by a, so that the number in front of  x^2 is a perfect square (1):

  x^2 + \frac{b}{a}x + \frac{c}{a} = {a}

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  \left(\frac{b}{2a}\right)^2 to both sides:

 \frac{y}{a} + \left(\frac{b}{2a}\right)^2 = x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 + \frac{c}{a}


3. Now notice that on the right, the first three terms factor into a perfect square:

 x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 = \left(x + \frac{b}{2a}\right)^2

Multiply this back out to convince yourself that this works.

4. Therefore the completed square form of the quadratic is:

 \frac{y}{a} + \left(\frac{b}{2a}\right)^2 = \left(x + \frac{b}{2a}\right)^2 + \frac{c}{a} 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.

2x2 + 24x + 23 = 0 Does not factor easily, so we complete the square.
x2 + 12x + 23/2 = 0 Make coefficient of x2 a 1, by dividing all terms by 2.
x2 + 12x = - 23/2 Add – 23/2 to both sides.
x2 + 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.