Section 11.8 Complex Numbers I. The Complex Number system A. The number i = -1 1. 9 and 24 B. Complex number have a Real part and an Imaginary part II. Powers of i 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i A. Cycle through 1. i =i, i 2 = -1, i 3 = - i, i 4 = 1 so Find i 5 ; i 18 ; and i 28 III. Addition and Subtraction A. Combine like terms ( 3 + 2i) + ( 5 7i) IV. Multiplication A. a b (ab) B. so write as complex numbers and then multiply 1. i i = -1 2. 2 14 C. Foil Complex numbers 1. (2+5i)(2-7i) D. Conjugate Pairs are the halves of the Difference of Squares form 1. (a+bi) (a-bi) = a 2 +b 2 2. (5-4i)(5+4i)
V. Dividing by a complex Number A. Rationalize the denominator 1. Single term denominators 2. two term denominators
Section 11.7 Solving Radical Equations I. Review Solving Equations A. Identify the type of Equation 1. Linear Equation 3x + 5 = 20 2. Quadratic Equation by factoring x 2 +2x = 24 3. Rational Equation 2 + x 3 5 = x 3 x 4 2 9 II. New Tool: Principle of Powers A. You usually maintain the equality of an equation when you raise both side of an equation to the same power 1. x 3 = 7 B. Raising both sides to an even power may not produce equivalent equations so you must check 1. x + 2 = 5 III. Use the Principle of Powers to Solve Radical Equations A. Steps 1. Isolate Radical 2. Raise both sides to a power equal to the index of the radical to remove the radical 3. Solve the resulting linear or quadratic equation 4. CHECK in original equation Some solutions maybe extraneous
B. Examples 1. 2 x 6 5 = 9 3 2. other powers x + 2 5 = 1 3. fractional exponents ( x + 2) 3 = 4 1 4. Binomial example 3 + 5 x = x a) Notice that this is a FOIL product b) checks are required IV. Solving Equation with two radicals expressions A. Must have one radical on each side of the equation B. Easy equation only two radicals, nothing else 1. 5t + 2 = t 12 C. Messy equations two radicals and other stuff 1. 6 x + 7 3x + 3 = 1 D. Steps for solving 1. Isolate one of the radicals 2. Use principle of powers to remove that radical sign a) Note that you must multiply out (FOIL) the other side and will still have a radical there 3. Isolate remaining radical 4. Use Principle of Powers again 5. Solve resulting equation 6. CHECK solutions do not always work
Section 12.1 Quadratic Equations V. Solving Quadratic Equations A. Identify 1. x 2 and = B. Solving By factoring x 2 + 3x = 28 1. Limitations; Only works for a small number of equations VI. Solving by Extracting the root (square root Property) A. Apply Principal of Powers to Square Roots B. Steps 1. Need both + and roots as answers 1. Can only use when perfect square = number 2. Isolate the perfect square 3. Take square root of both sides 4. Be sure to consider both + and roots 5. Check in equation and if both solutions make sense for the situation C. Examples 1. x 2 = 12 2. (x-2) 2 18 = 0 3. ((x+5) 2 + 16 = 0 D. Compound Interest (2 years annual) A = P(1 +r) t 1. Find the interest rate is after two years an account that is compounded annually has grown form $10,000 to $10,120
VII. Solve by Completing the Square A. To solve by completing the square you need to create a perfect square trinomial on one side of the equation so that you can factor your problem into a square = number and then extract the root B. Review Form for a perfect Square trinomial C. Steps 1. (A + B) 2 = A 2 + 2AB + B 2 2. (x-3) 2 = x 2 6x + 9 3. x 2 + 12x + would be a perfect square 1. x 2 and x terms on one side numbers on the other 2. Add same number to both side ( x coefficient / 2 squared) that will give you a perfect square trinomial on the x term side 3. Factor to perfect square then extract square roots 4. Check D. Examples: Solve 1. x 2 + 12x + 3 = 0 2. x 2 7x = 4 3. 5x 2 10x +3 = 0 4. 2x 2 + 7x = 8 2 E. Find the x intercepts of f ( x) = 3x + 2x 15 write as points
Review complete the square and order of operation Section 12.2 Quadratic Formula I. Quadratic Formula A. Development using Process of Complete the Square 1. On page 833 B. Formula finds two solutions to any quadratic =0 1. x = b ± b 2 4ac 2a 2. Major Benefit is that it always works 3. May return Complex answer (square root of negative) a) In that case also note NO REAL Solution 4. Find reduced radical form AND Use Calculator to approximate C. Examples II. The Discriminant A. Types of solutions with examples, 1. Type of Solutions depends on the Radical also called the Discriminant 2. Two Real x 2 + 8x 4 = 0 a) rational 6x 2 + x 35 3. One repeated Real 4x 2 2x +9 = 10x 4. Two imaginary, NO REAL Solutions 2x 2 5x + 10 = 0
III. Applications A. Pythagorean theorem 2 B. Falling Object Problems s = 4.9t + V0t + s0 1. Without initial velocity: A rock is dropped form the top of a cliff that is 40m high. How long before it hits the ground at the base of the cliff? 2. With initial velocity: A rock is thrown upward from the top of a cliff that is 40m high with an initial velocity of 10m/s. How long before it hits the ground at the base of the cliff 3. A rock is thrown downward from the top of a cliff that is 40m high with an initial velocity of 10m/s. How long before it hits the ground at the base of the cliff? C. X intercepts 1. axis of symmetry
Section 12.3 Equations in Quadratic form I. Solving Equations using substitution A. Is the exponent double? B. Examples II. Equations that become quadratic A. Polynomials that factor B. Rational Equations III. Writing Factors from rational solutions A. Find the solution to these factors =0 1. (2x + 3) 2. (x - 5) 3. (7x 4) B. For a solution of A/B the factor is (Bx A) C. Examples write the factors that are associated with these solutions 1. 2/5 2. 3 3. -7/3
IV. Finding one possible equation from Solutions A. Write Factors and multiply together to find an equation B. Only one of many 1. There may be a constant common factor which does not have a solution associated with it C. Examples: Write an equation that would have these solutions 1. 5 and -2/3 2. -2 and 5/4 3. 8, 6, -1
V. X intercepts of a polynomial function A. Solutions to the f(x) = 0 B. Each x intercept can be used to write a factor. C. Write an equation that has these x intercepts 1. (-3,0); (5/2, 0) 2. Use the graph below to write an equation