PRECALC ALG & TRIG
PRECALC ALG & TRIG MAC 1147
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This 10 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Larissa Williamson in Fall. Since its upload, it has received 21 views. For similar materials see /class/207048/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.
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Date Created: 09/19/15
L7 Complex Numbers and Quadratic Equations Complex Numbers The number i is a solution of the equation x2 1 that is i2 1 Another solution of this equation is i The imaginary number i is called the imaginary unit Complex numbers are numbers of the form a bi where a and b are real numbers The real numbers a and b are called respectively the real part and imaginary part of the complex number a bi The form a ib is the standard form of a complex number A real number a can be written as a i0 thus the set of real numbers is a subset in the complex number system Complex number bi is called a pure imaginary number Equality of Complex Numbers aibcia ifand only if 61 Sum of Complex Numbers aibcz d Product of Complex Numbers a 1bc id Note The product of complex numbers is easy to find by using FOIL and the fact that i2 1 Complex Conjugates The complex conjugate of a number Z a bi is 7 a bi Example Write expressions in the standard form z7abia bi z 7abi a bi 27 a bia bl 2701292 62 Quotient of Complex Numbers When evaluating the quotient of two complex numbers abl dz39 the conjugate of the denominator and simplify multiply both the numerator and denominator by Example Write in the standard form 1 l39 1i Powers of 139 1391 1392 1 3 1394 1 52 Example Evaluate 34 l 63 The Principle Square Root of a Negative Number If N gt 0 then J N AW is called the principle square root of N Note 1 i is the principle square root of 1 since J Ni 1 Another square root of l is l39 Caution When working with square roots of negative numbers first rewrite the root according to the definition above then apply any other rule for radicals 55 m Example Find the real numbers a and b so that the equation is true a62bl 4 5i 64 Example Write in standard form 3 l l 43 4146 Example Perform the operations and write the result in standard form 12 3z 7 5139 J3 2 705 2139 3 203 65 7 1 7 i Quadratic Equations with a Negative Discriminant The quadratic equation ax2 bx c O a 72 O has solutions b i W92 4016 201 If the discriminant 2 4016 lt 0 then the quadratic equation has two complex conjugate roots x 66 Example Use the quadratic formula to solve the equation in the complex number system x2 6x 10 2 0 Example A ball was thrown directly upward from the roof of a building After I seconds its height above the ground in feet is given by st l6t2 40t 24 Will the ball reach the top of a tree which is 56 foot tall 67 Example Solve for x 2x2 3xyy2 2 y 68 Equations that Lead to Quadratic Equations Example Solve the equation 1 8 x 2 x 2x x2 x 2 69 Example Rewrite without negative exponents and solve for x x 2 3x13O 70
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