Quadratic Equations

It is not necessary to extend our number field from the Real to the Complex at this juncture; complex numbers are two-dimensional, they require two generators to produce; however, it is perfectly legal to think of the reals as being embedded in the complex when the coefficient of the imaginary part is zero. This is similar to thinking of the integers as embedded in the rationals of the form, p/q, when either "p = q, or q = 1." The respective structures are the same; they are isomorphic.

Solving quadratic equations with one unknown:

Before we venture into the mechanics of method, let's analyze this equation from a geometric perspective. We are dealing with "one dimension" here, not in the sense of "linear," of course, but rather considering the fact of one unknown. Given a coordinate system, regardless of dimensionality, we will be focusing on the real number line, our "x" axis, with all other axes retracted to value zero.

The solution set must be such as to take us, stepwise, to the center of our axes system, or zero. Dividing by the coefficient of x², we may rewrite this equation in the form:

Let's consider, "+ d." This value is mapped through the function "ax²" to a point on our real line desgnated "ad²." It is then mapped through the function "- bx" to the point "- bd." And from here we add "- c" to arrive at the center of our system, zero. Zero is thus equal to the value "ad² - bd -c" which is "d" pushed through or mapped by "ax² - bx -c." This is fairly obvious; but what exactly is going on?

Let's look at an example to try to get an idea:

Considering "- 2," this value is mapped through x² to the point on our line corresponding to + 4. That is to say, we begin at the center, and "walk" four steps to the position numbered + 4. Now, - 2 is mapped through 3x to - 6. We turn again and walk six units to the point - 2. From here we add + 2; we turn and walk two units to the center of axes intersection, zero. These sub-functions act like direction vectors that will get us to the center if and only if the correct value is used.

Considering "- 1," this value is likewise pushed through x² to 1. We "walk " to the position designated as "1" on our number line. The value - 1 is then mapped by 3x to - 3. We turn and walk three paces to - 2. From here we turn and walk two steps to zero once again. These two values, (- 2, - 1), therefore represent the solution set of our equation, x² + 3x + 2.

If we were to graph a quadratic equation with 'real roots,' we would necessarily have to enter the second dimension, the plane. But as far as finding the solution set, it is not required to leave the land of one-dimension. The members of this set lie on the real number line or axis of our system.

Quadratic equations with Imaginary Roots are two-dimensional, the Complex number system being isomorphic to the Cartesian plane of Real number pairs.

Fundamentally there are three methods for arriving at the solution of a quadratic equation. They are:

• Factorization
• Completing the Square
• Using the Quadratic Formula

Factorization

The coefficient of the x² term has been divided through on both sides of the equal sign. This having been done, what we first concentrate on is the constant term, "36." Question: What whole number -- integer -- factors will yield this number upon mutiplication regardless of sign? We create a table:

This list constitutes all the integer factors of "36." Now, we examine both the sign, +/-, and the value of the "x" coefficient. We see that the sign is positive, and the value is "20." The sign tells us we must add our chosen factors. The only values which when added yield "20" are obviously "18" and "2." Therefore, the quadratic in our example factors to:

Our solution set, those values which will make the statement true are, (-18, -2). In general, once we have found the factors of the constant term, we have likewise found the solution set of our equation. Let's look at another example:

[RULE: multiplying like signs yields a positive;
multiplying unlike signs, a negative]

Once again we assume the coefficient of the "x²" term has been divided through. Our table for the constant is identical to the one above. We now examine the sign and value of the middle or "x" term. Because of the presence of both a positive and a negative sign, our question becomes twofold, or conditional:

What values which when added together will yield "-20";
what values which when multiplied together, "+36"?

Again the obvious answer is "-18" and "-2." Therefore our quadratic factors to:

Let's go on with another example to ingrain this simple process:

Again we establish a table for the constant term:

This table constitutes all the whole number combinations for "200." We must examine both the sign of the "x" term and that of the constant to discover more information. The sign of the "x" term is positive and that of the constant is negative. So we have a conditional question to ask at this point:

The only number pair having this dual property is, (20, -10).
Therefore, our quadratic factors to:

Completing the Square

Consider the general quadratic equation: ax² + bx + c = 0. Now, the basic idea with this method is to manufacture a perfect square on one side of the equal sign. And the way that's done is the following:

Let's try an example:

The Quadratic Formula

As regard the three possible choices of solutions:

Table of Contents
Equations of One Dimension || Linear Systems