Friday, November 2, 2012

Complex Roots (Also quadratic factoring)

Although I am far past the initial idea of complex roots in my mathematical education, it seems correct to explain it here, since it is the name of my blog.

 The first time I consciously remember a teacher saying the phrase, "What are the roots of this equation?" I was in a Trigonometry class my first semester at Seattle Central Community College. I strained my brain, but I seriously could not remember coming across this term earlier in my high school (which included Algebra 1 & 2, Geometry and Calculus). That night I had go teach the concept to myself (since I was too embarrassed to ask in class, which is a great thing to learn to overcome).


The term root is used in reference to a function.
The roots of a function, f(x), are the values of x that will cause the function to be zero.


a. f(x) = 2x

This equation has one root. That root is x = 0. No other value of x will cause f(x) to equal zero.

b. g(x) = (x² - 16)

There are two ways you can think of solving this. The first thing that I see when I look at this is a difference of squares. Difference of squares can be factored every time using this method.

(x²- c²) = (x + c)(x - c)

Using that method in this equation we have

g(x) = (x² -16)
        = (x + 4) (x - 4)

At this point we can see that if either (x + 4) = 0 or (x - 4) = 0 then the whole function will equal zero due to the Zero Product Property (any number multiplied by zero will equal zero).

The roots of this equation are therefore x = -4 and x =  4.

The other way we could have done this problem was not by factoring but just considering the problem (x² - 16). What values of x would give us 16? What numbers squared equal 16?  4 and -4.

Quadratic Formula

Not all equations can be factored using simple algebraic methods, such as the difference of squares method. When it is not obvious what the factors are, or you cannot find a method to easy factor the equation, it becomes necessary to use the quadratic equation.

The term quadratic equation, refers to an equation with one variable (x, usually) with a degree of two (meaning, the largest exponent in the equation is 2).  The basic form for a quadratic equation is

ax²+ bx + c = 0

A quadratic equation will always have two roots. 

Where x is the variable and a, b, and c all represent constants. To use the quadratic formula, you must put the equation in this form. Once the equation is in this form, you plug the values from that equation into the quadratic equation.

(-b ± √ b² - 4ac ) / 2a .

(Full disclosure: For some reason, the quadratic equation was one of the gaps in my education. I still have hard time recalling it from memory.  I seriously had to double check it for this post. Don't be like me, just pound it into memory the first time you learn it.)


c. Factor f(x) = x² + 5x - 2 using the quadratic equation.

The value for a, b and c are  a = 1, b = 5 and c = -2. We simply just plug these into the quadratic formula.

[-5 ± √ 5² - 4(1)( -2)] / 2(1) 

[-5 ± √41] / 2

The roots for this equations are x =  [-5 + √41] / 2 and [-5 - √41] / 2.

Although these numbers look "complex" they are not actually complex numbers. They are two real numbers, real roots to the equation.

(Fun fact: There is also a cubic formula and a quartic formula, a general equation for solving for the roots of a third degree or fourth degree polynomial. The cubic formula is rather big, but still digestible, and the quartic formula is ridiculously huge.  There actually can be no quintic or higher general function, however. This was proven in the 1800s. The proof for it is called Abel's Impossibility Theorem).

Complex Roots

A complex number is a number that includes the imaginary unit. The imaginary unit is  (√-1). The imaginary unit is represented by a lower case i.  The imaginary unit is literally defined as  i² = - 1 All complex numbers can be written in the form a+bi where a and b are constants. a is referred to as the real part of the number and bi is referred to as the imaginary part of the number. The idea of an imaginary number, intuitively, is difficult to grasp. I will not attempt to explain it here (mostly because I have problems grasping it myself).

Some quadratic equations, when put into a quadratic formula, give complex roots instead of real roots. This happens when the 4ac term of (√ b² - 4ac) is larger than the term.

( b² - 4ac is also referred to as the discriminate of the quadratic equation. A quadratic equation can have either one real root, two real roots, or two complex roots. If the discriminate = 0, then the equation will have one real root, if the discriminate > 0 the equation will have two real roots, if the discriminate < 0 it will have two complex roots. This can be useful to get an idea of what you are working with before you star the problem).


d. Factor f(x) = x² + 2x + 10 using the quadratic equation.

The quadratic equation is (-b ± √ b² - 4ac ) / 2a.

a = 1, b = 2, c = 10

[-2 ± √((2)² - 4(1)(10)] /  2(1)

[-2 ± √4- 40] /  2

You can tell that we will have complex roots based on the fact that the discriminate will be a negative number. 

[-2 ± √-36] /  2
[-2 ± (√36)(√-1)] /  2
[-2 ± 6(√-1) / 2
[-2 ± 6i] / 2 

-1 ± 3i 

The roots of this equation are x = -1 + 3i and x = -1 - 3i.