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5 Manipulate the following equation so it has the square of a binomial on the left side of the equals sign and 0 on the right side Here s a hint: The coefficient and constant in the binomial are both integers 16x 2 40x + 25 = 0 6 What is the single root of the quadratic stated in Prob 5 What is the solution set 7 Manipulate the following equation so it has the square of a binomial on the left side of the equals sign and a positive real number on the right side Here s a hint: The coefficient and constant in the binomial are both integers x2 + 6x 7 = 0 8 What are the roots of the quadratic stated in Prob 7 What is the solution set 9 How many real roots does the following quadratic have Find the real root or roots, if any exist, using the quadratic formula What is the real-number solution set 2x 2 + 3x + 35 = 0 10 How many real roots does the following quadratic have Find the real root or roots, if any exist, using the quadratic formula What is the real-number solution set 4x 2 + x + 3 = 0

Now that we know a little bit about what to expect when the discriminant of a quadratic equation is negative, let s explore this territory in more detail A negative discriminant indicates that the roots of a quadratic are imaginary or complex

xe x dx = [xe x e x ] 3 + [xe x e x ]0 =

The quadratic formula, like basic addition facts and multiplication tables, is worth committing to memory (That s why I keep repeating it Here it is again!) When we have a quadratic equation of the form ax 2 + bx + c = 0 then the roots can be found using the formula x = [ b (b 2 4ac)1/2] / (2a)

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Square root of the discriminant The discriminant of a quadratic equation is equal to the square of the coefficient of x, minus 4 times the product of the coefficient of x 2 and the stand-alone constant In the quadratic formula as stated above, the discriminant d is

1 2

d = b 2 4ac When the coefficients and constant in a quadratic are real numbers, then the discriminant is always a real number If d > 0, then the square root of d can be either a positive real or its additive inverse If d = 0, then the square root of d is 0 If d < 0, then the square root of d can be either of two values, one positive imaginary and the other negative imaginary

The work of computer programmers or software engineers, as they are called in the computer industry, involves writing detailed sets of instructions according to the problem descriptions and specifications of the systems analysts These programs are made up of a series of logical steps that the machine must follow in order for the data to be processed and transformed into usable information The programmer may use any one of a number of programming languages COBOL, Java, C++, and C are among the languages most employers seek Other development tools widely used today are Microsoft Visual Basic, Oracle Developer 2000, Visual C++, and Power Builder Today, most large corporate systems still run on COBOL, one of the older programming languages Programmers should have familiarity with DB2, Oracle, Microsoft SQL Server, Microsoft Access, Sybase Adaptive Server, or other database software There are more than one thousand computer languages It is essential that a programmer have a thorough knowledge of the language or languages in which programs are written in the organization where he or she seeks employment Programmers must further possess knowledge of general programming techniques and general relationships between program and hardware features A strong orientation to detail is important in that something as small as a misplaced comma could cause system failure This characteristic becomes crucial as a programmer debugs his or her programs to ensure they are error-free Debugging usually entails making trial runs on the computer with sample data Programs may be written in a matter of hours or may require more than a year of work Thus programmers may work alone on small projects or in teams for larger projects After completion of a program, the programmer must prepare an instruction sheet for the data entry and computer operators who will enter the data into the computer and run the program The work of these operators will be described in 4 There are three types of programmers at work in most organizations: operating systems programmers, applications programmers, and maintenance programmers.

Let s derive the SI and PS forms of linear equations by looking at how their graphs behave generally Then we ll derive a standard form for a linear equation based on two known points

0 1

Known slope and y -intercept Imagine a line in Cartesian coordinates that has slope m and crosses the y axis at the point (0, b), as shown in Fig 15-6 If we move away from (0, b) on the line, the slope is always equal to the difference in the y value divided by the difference in the x value, or y / x

Equations from Graphs +y x=0+ x y=b+ y (x,y)

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