Complex Numbers and Equation Solving 1. Simple Equations 2. Compound Equations 3. Systems of Equations 4. Quadratic Equations 5. Determining Quadratic Equations

Complex Numbers and Equation Solving In this section we will consider the following: a) simple equations b) compound equations c) systems of equations d) quadratic equations Simple Equations: Goal: isolate the variable on either the left or right side of the equation and to solve for one positive variable. Example 1: This example reviews the basic rules for solving a simple equation 1. Remove brackets or parentheses that exist within the equation using the distributive property. Reminder: make sure the value outside the ( ) or [ ] is distributed to all terms within, with a special focus on any sign changes. 2. Isolate the terms containing the variable remembering that if terms are moved across the equal sign, the sign of the term must change. 3. Simplify both sides of the equation. 4. Solve for “1” positive variable by multiplying both sides by the reciprocal of the coefficient or in simple terms divide both sides by the coefficient of the variable

Example 2: Real coefficient assigned to the variable with real and imaginary numbers Example 3: Imaginary coefficients assigned to the variable with imaginary numbers 1. Isolate the terms containing the variable “x” on the left side of the equation and move all real and imaginary numbers to the right side. 2. Simplify both sides of the equation. 3. Solve for one positive “x” 1. Remove any parentheses. 2. Isolate the terms containing the variable “x” on the left side of the equation and move all imaginary numbers to the right side. 3. Simplify both sides of the equation. 4. Solve for one positive “x”. Dividing both sides of the equation by “-5i” results in a situation where an imaginary number is contained in both numerator and denominator and as a result the value “i” cancels.

Example 4: Imaginary coefficients assigned to the variable with real and imaginary numbers Example 5: Real and imaginary coefficients assigned to the variable with real numbers 1. Remove any parentheses. 2. Isolate terms containing the variable “x” on the left side and the real and imaginary numbers on the right side of the equation. 3. Simplify both sides of the equation 4. When solving for one positive “x” you must divide by an “i” and following the rule of no imaginary number in the denominator, both numerator and denominator must be multiplied by an “i”. 5. Simplify the resulting fraction the conjugate of the complex number factor (7 + 2i). 4. Simplify the resulting fraction 1. Isolate and simplify as in previous examples 2. Remove the common factor “x” from the left side of the equation resulting in a complex number factor (7 - 2i). 3. When solving for one positive “x” you must divide by the complex number factor and following the rule of no imaginary number in the denominator, the denominator and numerator on the right side must be multiplied by

Example 6: Real and imaginary coefficients assigned to the variable with imaginary numbers 1. Remove any parenthesis. 2. Isolate the variables on the left side and the imaginary numbers on the right side of the equation. 3. After simplifying the equation, remove the common factor “x” from the left side of the equation resulting in a complex number factor (-2i - 6). 4. When solving for one positive “x” you must divide by the complex number factor and following the rule of no imaginary number in the denominator, the denominator and numerator on the right side must be multiplied by (-2i + 6). 5. Simplify the resulting fraction. Some basic items to take into consideration: a) the distributive property b) the value of c) when reducing a fraction every term is affected by a common factor or a negative sign. In this example “-4 ”

Example 7: Real and imaginary coefficients assigned to the variable with real and imaginary numbers 1. Remove any parenthesis. 2. Isolate the variables on the left side and the real and imaginary numbers on the right side of the equation. 3. After simplifying the equation, remove the common factor “x” from the left side of the equation resulting in a complex number factor (-5i - 3). 4. When solving for one positive “x” you must divide by the complex number factor and following the rule of no imaginary number in the denominator, the denominator and numerator on the right side must be multiplied by (-5i + 3). 5. Simplify the resulting fraction. Some basic items to consider: a) binomial expansion b) the value of c) when reducing a fraction every term is affected by a common factor or a negative sign. In this example “a negative sign”.

Compound Equations By definition - an equation containing two different variables - one with real coefficients assigned and the second having assigned imaginary coefficients. Goal - to create two equations by isolating the real equation from the imaginary and solving each equation for each particular variable following the procedures outlined for simple equations. (Check procedure by referring to identified example) Example 1: Therefore, the complex number on the left side = the complex number on the right Solution Set Check: The real equation formed by equating the real values from the left with the real values on the right side of the equal sign. (Check example 1) Real Equation Imaginary Equation The imaginary equation formed by equating the imaginary values from the left with the imaginary values on the right side of the equal sign (Check example 3)

Example 2 The real equation formed by equating the real values from the left with the real values on the right side of the equal sign. (Check example 1) The imaginary equation formed by equating the imaginary values from the left with the imaginary values on the right side of the equal sign (Check example 3) Therefore, the complex number on the left side = the complex number on the right Solution Set {(-1, 6)} Check Imaginary Equation Real Equation Remember to remove parenthesis

System of Equations By definition - an equation containing two different variables - each with real and imaginary coefficients assigned. Goal - a) to create two equations by isolating the real equation from the imaginary following procedure of real values equal real values and imaginary values equal imaginary values b) to simplify each equation using procedures of simple equations c) to solve the two equations as a system using the procedures of elimination or substitution. Review procedures by referring to Solving Systems of Equations Example 1 Imaginary Equation Real Equation After each equation has been formed, each is simplified and put into the form of Ax + By = C Solving the elimination - removing the x and then solving for one positive ‘y” Solving for “y” Substitution of The system of equations

Check Therefore, the complex number on the left side = the complex number on the right Solution Set The next example will use the process of substitution to solve the system of equations for “y” and solving for “x”

Example 2 After each equation has been formed, each is simplified and put into the form of Ax + By = C Real Equation Imaginary Equation Solving for “x” Substitution of for “x” and solving for one positive “y” Substitution of for “y” and solving for one positive “x” The system of equations

Check Therefore, the complex number on the left side = the complex number on the right Solution Set Quadratic Equations The previous examples contained a visual “i” and it was easy to identify each equation as one that contained imaginary or complex numbers. The quadratic equation with real coefficients hides any reference to the existence of imaginary or complex numbers until one is asked to solve (determine the roots, find the critical zeros, calculate the x-intercepts) the equation. The procedure used most often is the application of the quadratic formula: and in particular, the discriminant. If the value of the discriminant is less than zero (a negative value) the nature of the roots can be interpreted as two unequal and imaginary

roots. A further interpretation is that a graph of this function would not have any x- intercepts or critical zeros. One other note is that imaginary roots always occur in pairs and are conjugates on one another. Example 1: Once the roots have been determined one task remains and that is to check whether the calculated roots are the correct values. Two distinct procedures can be used. a) The first procedure requires the substitution of a root into the original equation, simplification and a result of 0 = 0. b) The second procedure utilizes two parts i) the formulas -b/a and c/a which equals the sum and the product of the roots respectively. ii) the addition of and multiplication of the calculated roots iii) comparison of the results from i) and ii) Note: 1. Take time when writing out and substituting values into the formula paying particular attention to negative signs. 2. Use the concept of to remove the negative values from under the radical sign 3. Remove all perfect from under the radical.

Procedure One Procedure Two Note: 1. Adding conjugates results in the radical term being eliminated. 2. When multiplying conjugates the middle term is always eliminated. 3. A radical times itself always results in an answer of the radicand. To help simplify the equation, remove the fractions be multiplying each term by a common denominator.Remember to use the distributive property on the second fraction. Solution Set:

Example #2: Note: Always pay attention to the substitution of negative values for a, b and c Check: Be careful with sign changes when simplifying this part of the equation

Check 2: Solution Set Determining a Quadratic Equation The ability to calculate the roots of a quadratic equation also allows for the mathematical procedure of determining a quadratic equation given the roots of an equation. Two procedures can be used. The include: a) writing the equation as a product of factors and then simplifying the equation b) using the formula The following examples will demonstrate the two procedures. Evaluate both procedures and decide which one is best suits your mathematical skills.

Example 1: non fractional roots When the roots are not fractions the amount of work required does not appear to be that much different but additional work is still required when using binomial expansion in the first procedure as identified by the red boxes.

Example 2: fractional roots This example demonstrates that the first procedure requires a lot more work especially when the binomial expansion involves fractions as shown by the red boxes. The last part of both procedures requires the same amount of calculation

Example 3: the roots written as a combined expression If not indicated, the choice of which procedure (factors or formula) is left up to you. Make your choice based on mathematical skill and chance for best success in determining the equation To determine the equation it is necessary to identify the given 2 roots

A special application of imaginary numbers A quick focus on factoring: Each of these examples represent a difference of squares and the factors are a set conjugates formed by taking the square root of each of the values in the question. Until now questions of the form have not been factorable. The existence of imaginary numbers permits the factoring of an equation because of this basic fact which in term allows positive perfect squares to be re-written as a product as indicated in these examples To factor questions of the form Step 1: re-write the question so that appears as a difference of squares (this is accomplished by using the above fact) Include binomial multiplkicatiom