DrDelMath

Chapter 11: Quadratic Equations

Section 11.1 and Section 11.2: Quadratic Equations in One Variable (Page 651)

To say an equation is written in the form $a{x}^2+bx+c=0$ where a, b, and c are real numbers and $a\ne 0$ means that the equation has a second degree polynomial on one side of the equality and the number 0 on the other side. The only restrictions on the coefficients a, b, and c is that they be real numbers and $a\ne 0$. Observe that permits the coefficients to be natural numbers, whole numbers, integers, rational numbers, or irrational numbers. When a quadratic equation is written in this form it is said to be in standard form. The standard form is important to us because all methods to solve quadratic equations in one variable are based on the assumption that the equation is already written in standard form.

If it is desired to solve a quadratic equation in one variable which is not written in standard form, one must write it in standard form by using only methods which generate equivalent equations, before attempting to use any of the methods to solve a quadratic equation in one variable. It is important that the rewitten equation be equivalent to the original equation otherwise its solutions will not be solutions of the original equation. Generally one or both of the following two methods are enough to transform an equation into standard form.

Take a close look at the statement of The Zero Factor Property. There is no mention of quadratic equations. The only mathematics objects mentioned are real numbers. The Zero Factor Property is a property of the Real Number System. It is used in many different situations. Any time a product of real numbers (or expressions representing real numbers) is zero, then at least one of the factors is zero. All too often the student only sees this property used for solving quadratic equations. In fact it can be used to answer all kinds of questions like:

• Solve a quadratic equation like $(x+2)(x+5)=0$


• Solve a higher degree polynomial equation like $\left(3x-5\right)\left(x+2\right)\left(5x-6\right)\left(x-1\right)=0$
  


• Determine the domain of a rational expression like ${\displaystyle \frac{(4x-5)(x+2)(x-8)}{x(x+3)(x-2)}}$
  


• Determine the values of x which cause a rational expression like ${\displaystyle \frac{(4x-5)(x+2)(x-8)}{x(x+3)(x-2)}}$ to be zero.
  

Quadratic equations may be solved by factoring in conjunction with the Zero Factor Property or by using the Quadratic formula.

Quadratic Equations: Two Most Common Methods for Solving

This is a good point to pause for a review of some elementary facts and properties of Complex Numbers.
A detailed presentation of Complex Numbers is found HERE

Quadratic Equations: Two Less Used Methods for Solving

The Square Root Property

Solving a quadratic equation using the Square Root Property depends on the definition of square root of a nonnegative number.

Suppose we want to solve the equation $x^2=5$. To solve that equation means to find a number $x$ whose square is 5. According to the above definition the square root of 5 is a number whose square is 5. Note that the square of both the Principal Square Root and the negative square root make the equation true. Therefore the solutions are $\pm \sqrt{5}$ and the solution set is $\left\{\pm \sqrt{5}\right\}$

A slight variation of the above problem might be the solution of ${\left(x-3\right)}^2=16$.

$\begin{array}{l}{\left(x-3\right)}^2=16\\ x-3=\pm \sqrt{16}=\pm 4\\ x=3\pm 4\\ \text{The solution set is }\left\{-1,7\right\}\end{array}$

Solving a quadratic equation with the Square Root Property is an application of the definition of Square Root and is useful in those very special occasions when the equation consists of a square equal to a number.

Completing The Square

The process of completing the square is a very useful tool which is used in many situations in mathematics. In fact it is the best approach to derive the Quadratic Formula. This derivation is presented on Page 662 at the beginning of Section 11.2.
One could say completing the square was used to develop the quadratic formula to avoid continued use of completing the square to solve quadratic equations.
Completing the square is not a sensible method for solving quadratic equations.

Section 11.3: Using Quadratic Methods (Substitution) to Solve Certain Special Equations

The material under this topic is normally presented with very little fanfare as a dull mechanical process. However, an examination below the surface quickly reveals two very general and very powerful problem solving techniques.

• Convert a new problem to an old familiar problem.
• Substitution of variables to obscure the unimportant peculiarities of the problem and highlight the important details.

Both of these techniques are used extensively in mathematics and are also used in many non-math problem resolution scenarios.
When first confronted with the task of solving a quadratic equation in one variable, it is pretty natural to use factorization and The Zero Factor Property to convert that problem to the problem of solving linear equations in one variable -- a familiar easily solved problem.
When faced with an application problem (word problem) it is imperative to substitute variables for the important terms during the process of creating a mathematical model. The model is "divorced" from the confusing language of the problem.

In this section we are going to solve equations which are "Quadratic in Form". Quadratic in Form simply means that substitution of a single variable for some expression in the equation produces a quadratic equation. Because we are expert at solving quadratic equations, the substitution converts a new problem to a familiar problem. At this point in the process we feel confident we can solve the problem -- just a matter of cranking out those pesky numbers.

An equation is quadratic in form if it looks like $a{M}^2+bM+c=0$ where $a\ne 0,b,c$ are numerical coefficients and M is some algebraic expression.

When you are faced with a new equation in one variable, try to rewrite (using the First Two Properties of Equations) the equation in a more friendly/yielding format. If that does not work, check to see if it is Quadratic in Form. If it is, use the techniques of this section to solve the equation. For some equations the person solving the equation must be a creative, ingenious artist.

Section 11.5 and 11.6: Quadratic Functions and Parabolas

Function: Quadratic

Important: In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P.
As with many words in mathematics we are much more precise when we use the word characterize.
Consider the statement: "Points on the x-axis are characterized by the fact that their second coordinates are 0."
We mean that the second coordinate of every point on the x-axis is 0 and points not on the x-axis do not have second coordinates 0.
So if we are "looking for points on the x-axis we simply "algebraically run around looking" for points whose second coordinates are 0.

"Points on the y-axis are characterized by the fact that their first coordinates are 0."
We mean that the first coordinate of every point on the y-axis is 0 and points not on the y-axis do not have first coordinates 0.
So if we are "looking for points on the y-axis we simply "algebraically run around looking" for points whose first coordinates are 0.

Determing Intercepts of Graphs of Functions Recall that the graph of a function named $f$ is the set of points of the form $\left(x,f\left(x\right)\right)$. That is -- The second coordinate is the range element associated with the first coordinate.

y-intercept: To find the y-intercept of a function $f$, we must realize two important properties of the point we are trying to find:

• the point is on the graph of $f$ so has the form $\left(x,f\left(x\right)\right)$.
• The point is on the y-axis so its first coodinate must be 0. So the point must be of the form $\left(0,y\right)$.

x-intercept: To find the x-intercept of a function $f$, we must realize two important properties of the point we are trying to find:

• the point is on the graph of $f$ so has the form $\left(x,f\left(x\right)\right)$.
• The point is on the x-axis so its second coodinate must be 0. So the point must be of the form $\left(\mathrm{x},0\right)$.

INSERT EXAMPLES OF COMPLETED GRAPHS

We "sketch" the graph of a function and do not expect the graph to be precise or to scale.
The apparent coordinates on a graph of a function should be considered to be only approximations (sometimes very crude approximations). Points whose coodinates appear on the graph are the only points whose coordinates may be assumed to be accurate.

Any of the above graphs of quadratic functions illustrate most of the important features that a graph of a function should show.

• The general shape
• The orientation
• End behavior
• The domain
• The range
• The high and low points (relative to their neighboring points)
• Where (for what values of x) is the graph on the x-axis -- where is f(x) = 0 -- The x-intercepts.
• Where (for what values of x) is the graph above the x-axis -- where is f(x) > 0
• Where (for what values of x) is the graph below the x-axis -- where is f(x) < 0

Section 13.1: Circles

Formulas

Completing the Square Examples
Exercises Related to Circles
Solutions for Exercises