College Algebra 9th txtbk

84 CHAPTER 1 EQUATIONS AND INEQUALITIES
Letters z and w are often used as complex variables, where
z x yi, w u vi, and x, y, u, v are real numbers. The
conjugates of z and w, denoted by z and w, respectively, are given
by z x yi and w u vi. In Problems 91–98, express each
property of conjugates verbally and then prove the property.
91. zz is a real number. 92. z z is a real number.
93. z z if and only if z is real. 94. z z
95. z w z w
96. z w z w
97. zw z w
98. zw z w
1-5 Quadratic Equations and Applications
Z Using Factoring to Solve Quadratic Equations
Z Using the Square Root Property to Solve Quadratic Equations
Z Using Completing the Square to Solve Quadratic Equations
Z Using the Quadratic Formula to Solve Quadratic Equations
Z Solving Applications Involving Quadratic Equations
The next class of equations we consider are the second-degree polynomial equations in one
variable, called quadratic equations.
Z DEFINITION 1 Quadratic Equation
A quadratic equation in one variable is any equation that can be written in the
form
ax 2 bx c 0 a 0
where x is a variable and a, b, and c are constants.
Standard Form
Now that we have discussed the complex number system, we can use complex numbers
when solving equations. Recall that a solution of an equation is also called a root of
the equation. A real number solution of an equation is called a real root, and an imaginary
number solution is called an imaginary root. In this section, we develop methods for finding
all real and imaginary roots of a quadratic equation.
Z Using Factoring to Solve Quadratic Equations
There is one single reason why factoring is so important in solving equations. It’s called
the zero product property.
ZZZ EXPLORE-DISCUSS 1
(A) Write down a pair of numbers whose product is zero. Is one of them zero?
Can you think of two nonzero numbers whose product is zero?
(B) Choose any number other than zero and call it a. Write down a pair of numbers
whose product is a. Is one of them a? Can you think of a pair, neither of which is
a, whose product is a?