CHAPTER 7 - QUADRATIC EQUATIONS and INEQUALITIES (Pages 325 to 378) In this chapter the real number system will be expanded to the complex number system, and the general solution to the second degree equation in one variable will be developed.
LESSON #7.1 - The General objective of this lesson is to expand the real number system to the complex number system.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to do arithmetic using complex numbers.
WORD PROBLEMS:
7.1 - Complex Numbers - In Chapter One the real number system was introduced, and although many operations can be performed with these numbers, some operations are excluded. Specifically even roots of negative numbers. This can be remedied with the following definition.
Definition of the Imaginary Unit i -
As a result of this definition we can write -
i2 = -1
i3 = -i
i4 = 1
This cycles as the subscript increases.
Square Root of a Negative Number - Any negative real number can be written as -
-y2
where y is a real number. Let us now consider the square root of this number.
This result shows us that associated with every real number 'y', there is an imaginary number 'iy'.
Definition of the Complex Number z - A new number, called a complex number, is defined as -
z = x + iy
where x and y are real numbers, and i is the imaginary unit. Since any real number can be written as a complex number with y equal to zero, the set of complex numbers includes the set of real numbers.
Complex Number Arithmetic - Let z1 and z2 be two complex numbers expressed as -
z1 = x1 + iy1
z2 = x2 + iy2
Addition:
z1 + z2 = x1 + iy1 + x2 +iy2
z1 + z2 = (x1+x2) + i(y1+y2)
Subtraction:
z1 - z2 = x1 + iy1 - (x2 +iy2)
z1 - z2 = (x1-x2) + i(y1-y2)
Multiplication:
z1 x z2 = (x1 + iy1) x (x2 +iy2)
z1 x z2 = x1x2+ ix2y1 + ix1y2 + i2y1y2
z1 x z2 = (x1x2 - y1y2) + i(x2y1 + x1y2)
Complex Conjugates - Two complex numbers x + iy and x - iy are called conjugates of each other. Their product is -
(x + iy) x (x - iy) = x2- i2y2
(x + iy) x (x - iy) = x2 + y2
which is a real number. This property is used to divide complex numbers.
Division:
Board and Class Problems - Pages 332 to 333/16, 24, 70, 98.
LESSON #7.2 - The General objective of this lesson is to introduce the general quadratic equation.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to identify the standard form of a quadratic equation.
7.2 - Quadratic Equations - The standard form of the second-degree or quadratic equation is -
ax2 + bx + c = 0
where a, b and c are real numbers.
There are four basic techniques for solving such equations. They are -
Factoring - We have already seen how factoring can be used to solve some quadratic equations. If the left-hand side can be factored, the quadratic equation can be written as -
(dx + e)(fx + g) = 0
Setting each factor equal to zero yields the following solutions -
dx + e = 0 fx + g = 0
x = -e/d x = -g/f
Taking the Square Root of Both Sides - If the equation can be expressed as -
x2 = a
it can be solved by taking the square root of both sides; i.e.
Board and Class Problems- Pages 339 to 341/88, 90.
LESSON #7.3 - The General objective of this lesson is to learn how to solve quadratic equations by completing the square.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve any quadratic equation by completing the square.
7.3 - Completing the Square - This technique can be used to solve any quadratic equation. However, it is the most complicated technique. We will illustrate this technique by solving the following equation -
Board and Class Problems - Pages 345 to 347/58.
LESSON #7.4 - The General objective of this lesson is to learn the quadratic formula.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve any quadratic equation using the quadratic formula.
7.4 - Quadratic Formula - The technique of completing the square can be used to derive the quadratic formula, which can be used to solve any quadratic equation. Solving the following equation -
Board and Class Problems - Page 355/24.
Quadratic Discriminant - The expression
b2 - 4ac
is called the quadratic discriminant. It is called this, because it indicates the nature of the solutions to the quadratic equation.
Sum and Product of the Roots-
Sum - x1 + x2 = "-b/a"
LESSON #7.5 The General Objective of this lesson is to solve word problems involving quadratic equations.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve word problems involving quadratic equations.
WORD PROBLEM - In the lesson.
7.5 - More Quadratic Equations and Applications - Review the rules from Chapter 2 about solving word problems.
Rules for Solving Word Problems -
Step 1. Read the problem very carefully, making sure you understand what is stated and what is being asked for.
Step 2. If a diagram is appropriate, sketch it, and use it to solve the problem.
Step 3. Answer the following question.
What am I looking for?
Write down the answer, and then give the quantity(s) a mathematical name(s) such as x, y, etc.
Step 4. Write down any other quantities specified in the problem along with appropriate mathematical names. Use a table as an organizing tool when appropriate.
Step 5. Rewrite the problem in the form of an equation, focusing on keywords such as plus, is, equal to, difference of, etc.. (Refer to page 34 in the text.)
Step 6. Solve the equation for the unknown variable.
Step 7. Check the solution to see if it agrees with the facts in the problem. As an aid, answer the following question if you can.
Does the solution make sense?
If the answer to this question is yes, the odds are very good that you solved the problem correctly. If the answer is no, go back and resolve the problem.
Board and Class Problems Pages 363 to 365/46, 54, 58, 60, 62, 64.
LESSON #7.6 - The General objective of this lesson to show how quadratic inequalities can be solved.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to; Solve quadratic inequalities.
WORD PROBLEM - A pilot flew east against a headwind of 60 mph, and then made a return trip with a tailwind of 40 mph. The air speed of the plane was 360 mph, and the round trip required seven hours. How long and how far did he fly east?
7.6 Quadratic Inequalities - Quadratic inequalities can be solved using one of the following techniques.
Factoring - If the quadratic inequality is factorable you can write -
ax2 + bx + c > 0
(dx + e)(fx + g) > 0
This means that one of the following two conditions is correct -
i. (dx + e) > 0 and ( fx + g ) > 0
ii. (dx + e) < 0 and ( fx + g ) < 0
Each of the linear inequalities can now be solved. Each condition will yield half of the solution.
If the inequality has the form-
ax2 + bx + c < 0
(dx + e)(fx + g) < 0
This means that one of the following two conditions is correct
i. (dx + e)> 0 and (fx + g) <0
ii. (dx + e) < 0 and (fx + g) > 0
Each of the linear inequalities can now be solved. One of these conditions will yield the entire solution.
Board and Class Problems - Page 371/6.
Critical Points - Given a quadratic inequality -
ax2 + bx + c > 0
Solve the associated equality -
ax2 + bx + c = 0
This will yield two solutions, x1 and x2, that divides the number line into three regions as shown in Figure 7:1.
Figure 7:1 - Critical Points for Quadratic Inequalities
Discuss the roles of the three regions A, B and C. Illustrate how a test point can be used to determine which of these regions satisfies the given inequality.
Board and Class Problems - Solve the inequality x 2 + 3x + 1 > 0.
HOMEWORK - Lesson Seven Assignment
COMMENTS -