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Algebra

Explanation of Derivation

The section "arranged by culture" on the Wikipedia page List of numeral system topics gives a feel for the different counting systems different cultures have come up with. A little practice doing calculations with other ancient number systems like the Roman numerals will show the advantages of Hindu-Arabic numerals. Mathematics before the introduction of Hindu-Arabic numerals and our notation for variables was expressed very differently. Much of the history of math is the story of extending existing notations or creating new notations to express mathematical ideas. The history section on the Wikipedia page Quadratic equations shows that methods like the one described below were known as early as 2,000 BC, and were still being published in journals as let as 1896.

The method for completing the square started with a geometric representation, but is even more powerful for transforming equations symbolically. With practice you will find it easy to use for solving and graphing quadratic functions, but it is also necessary for transforming equations in later mathematics classes you will take.

Completing The Square

In this drawing we've divided area a into unit squares that are . The $x^{2}$ term groups these units into the largest square possible while the ${b}x$ term represents the remaining units to get to a. For instance if a=40 then x=6 and b=4/6 or if a=44 then x=6 and b= 8/6.

Divide the term in front of the x in half.

And create two strips on the side of your square.

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

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.

Derivation

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,

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