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College Algebra. Exam 2 Material. Quadratic Applications. Application problems may give rise to all types of equations, linear, quadratic and others Here we take a look at two that lead to quadratic equations. Example.
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College Algebra Exam 2 Material
Quadratic Applications • Application problems may give rise to all types of equations, linear, quadratic and others • Here we take a look at two that lead to quadratic equations
Example Two boys have two way radios with a range of 5 miles, how long can they communicate if they leave from the same point at the same time with one traveling north at 10 mph and the other traveling east at 7 mph? D = R T N boy E boy
Example • A rectangular piece of metal is 2 inches longer than it is wide. Four inch squares are cut from each corner to make a box with a volume of 32 cubic inches. What were the original dimensions of the metal? Unknowns L Rec W Rec L Box W Box .
Homework Problems • Section: 1.5 • Page: 130 • Problems: 5 – 9, 21 – 22 • MyMathLab Assignment 22 for practice
Other Types of Equations • Thus far techniques have been discussed for solving alllinear and quadratic equations and some higher degree equations • Now address techniques for identifying and solving many other types of equations
Solving Higher Degree Polynomial Equations • So far methods have been discussed for solving first and second degree polynomial equations • Higher degree polynomial equations may sometimes be solved using the “zero factor method” or, the “zero factor method” in combination with the “quadratic formula” or the “square root property” • Consider two examples:
Example One Make one side zero: Factor non-zero side: Apply zero factor property and solve:
Example Two x3 + x2 – 9x – 9 = 0 One side is already zero, so factor non-zero side x3 + x2 – 9x – 9 = 0 x2(x + 1) – 9(x + 1) = 0 (x + 1)(x2 – 9) = 0 Apply zero factor property and solve: x + 1 = 0 OR x2 – 9 = 0 x = -1 x2 = 9 x = ± 3
Homework Problems • Section: 1.4 • Page: 124 • Problems: 59 – 62 • MyMathLab Assignment 23 for practice
Rational Equations • Technical Definition: An equation that contains a rational expression • Practical Definition: An equation that has a variable in a denominator • Example:
Solving Rational Equations • Find“restricted values” for the equation by setting every denominator that contains a variable equal to zero and solving • Find the LCD of all the fractions and multiply both sides of equationby the LCD to eliminate fractions • Solve the resulting equation to find apparent solutions • Solutions are all apparent solutions that are not restricted
Homework Problems • Section: 1.6 • Page: 144 • Problems: Odd: 1 – 25 • MyMathLab Assignment 24 for practice
“Quadratic in Form” Equation • An equation is “quadratic in form” if the same algebraic expression is found twice where one time the exponent on the expression is twice as big as it is the other time • Examples: m6 – 7m3 – 8 = 0 8(x – 4)4 – 10(x – 4)2 + 3 = 0
Solving Equations that areQuadratic in Form • Make a substitution by letting “u” equal the repeated expression with exponent that is half of the other • Solve the resulting quadratic equation for “u” • Make a reverse substitution for “u” • Solve the resulting equation
Homework Problems • Section: 1.6 • Page: 145 • Problems: All: 61 – 64, 73 – 74 • MyMathLab Assignment 25 for practice
“Negative Integer Exponent” Equation • An equation is a “negative integer exponent equation” if it has a variable expression with a negative integer exponent • Examples:
Solving “Negative Integer Exponent” Equations • If the equation is “quadratic in form”, begin solution by that method • Otherwise, use the definition of negative exponent to convert the equation to a rational equation and solve by that method
Homework Problems • Section: 1.6 • Page: 145 • Problems: 75, 76 • MyMathLab Assignment 26 for practice • MyMathLab Homework Quiz 5/6 will be due for a grade on the date of our next class meeting
Radical Equations • An equation is called a radical equation if it contains a variable in a radicand • Examples:
Solving Radical Equations • Isolate ONE radical on one side of the equal sign • Raise both sides of equation topower necessary to eliminate the isolated radical • Solve the resulting equation to find “apparent solutions” • Apparent solutions will be actual solutionsif both sides of equation were raised to an odd power, BUTif both sides of equation were raised to an even power, apparent solutionsMUSTbechecked to see if they are actual solutions
Why Check When Both Sides are Raised to an Even Power? • Raising both sides of an equation to a power does not always result in equivalent equations • If both sides of equation are raised to an odd power, then resulting equations are equivalent • If both sides of equation are raised to an even power, then resulting equations are not equivalent (“extraneous solutions” may be introduced) • Raising both sides to an even power, may make a false statement true: • Raising both sides to an odd power never makes a false statement true: .
Homework Problems • Section: 1.6 • Page: 144 • Problems: Odd: 27 – 51, 55 – 57 • MyMathLab Assignment 27 for practice
Rational Exponent Equations • An equation in which a variable expression is raised to a “fractional power” Example:
SolvingRational Exponent Equations • If the equation is quadratic in form, solve that way • Otherwise, solve essentially like radical equations • Isolate ONE rational exponent expression • Raise both sides of equation to power necessaryto change the fractional exponent into an integer exponent • Solve the resulting equation to find “apparent solutions” • Apparent solutions will be actual solutions if both sides of equation were raised to an odd power, but if both sides of equation were raised to an even power, apparent solutions MUST be checked to see if they are actual solutions
Homework Problems • Section: 1.6 • Page: 145 • Problems: All: 53 – 54, 59 – 60, 65 – 72 • MyMathLab Assignment 28 for practice
Definition of Absolute Value • “Absolute value” means “distance away from zero” on a number line • Distance is always positive or zero • Absolute value is indicated by placing vertical parallel bars on either side of a number or expression Examples: The distance away from zero of -3 is shown as: The distance away from zero of 3 is shown as: The distance away from zero of u is shown as:
Absolute Value Equation • An equation that has a variable contained within absolute value symbols • Examples: | 2x – 3 | + 6 = 11 | x – 8 | – | 7x + 4 | = 0 | 3x | + 4 = 0
Solving Absolute Value Equations • Isolate one absolute value that contains an algebraic expression, | u | • If the other side is a negative number there is no solution (distance can’t be negative) • If the other side is zero, then write: u = 0 and Solve • If the other side is “positive n”, then write: u = n OR u = - n and Solve • If the other sideis another absolute value expression, | v |, then write: u = v OR u = - v and Solve
Homework Problems • Section: 1.8 • Page: 164 • Problems: Odd: 9 – 23, 41 – 43, 67 – 69 • MyMathLab Assignment 29 for practice • MyMathLab Homework Quiz 7 will be due for a grade on the date of our next class meeting
Inequalities • An equation is a comparison that says two algebraic expressionsare equal • An inequality is a comparison between two or three algebraic expressions using symbols for: greater than: greater than or equal to: less than: less than or equal to: • Examples: .
Inequalities • There are lots of different types ofinequalities, and each is solved in a special way • Inequalities are called equivalent if they have exactly the same solutions • Equivalent inequalities are obtained by using “properties of inequalities”
Properties of Inequalities • Adding or subtracting the same number to all parts of an inequality gives an equivalent inequalitywiththe samesense (direction) of the inequality symbol • Multiplying or dividing all parts of an inequality by the same POSITIVE number gives an equivalent inequalitywith the same sense (direction) of the inequality symbol • Multiplying or dividing all parts of an inequality by the same NEGATIVE number and changing the sense (direction) of the inequality symbol gives an equivalent inequality
Solutions to Inequalities • Whereas solutions to equations are usually sets of individual numbers, solutions to inequalities are typically intervals of numbers • Example: Solution to x = 3 is {3} Solution to x < 3 is every real number that is less than three • Solutions to inequalities may be expressed in: • Standard Notation • Graphical Notation • Interval Notation
Two Part Linear Inequalities • A two part linear inequality is one that looks the same as a linear equationexcept that an equal sign is replaced by inequality symbol (greater than, greater than or equal to, less than, or less than or equal to) • Example:
Expressing Solutions to Two Part Inequalities • “Standard notation” - variable appears alone onleft side of inequality symbol, and a number appears alone onright side: • “Graphical notation” - solutions are shaded on a number line using arrows to indicate all numbers to left or right of where shading ends, and using a parenthesis to indicate that a number is not included, and a squarebracket to indicate that a number is included • “Interval notation” - solutions are indicated by listing in order the smallest and largest numbers that are in the solution interval, separated by comma, enclosed within parenthesis and/or square bracket. If there is no limit in the negative direction, “negative infinity symbol” is used, and if there is no limit in the positive direction, a “positive infinity symbol” is used. When infinity symbols are used, they are alwaysused with a parenthesis.
SolvingTwo Part Linear Inequalities • Solve exactly like linear equationsEXCEPT: • Always isolate variable on left side of inequality • Correctly apply principles of inequalities (In particular, always remember to reverse sense of inequality when multiplying or dividing by a negative)
