PRECALC ALG & TRIG
PRECALC ALG & TRIG MAC 1147
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This 8 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 56 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
L5 Quadratic Equations and the Quadratic Formula Applications Complex Numbers Quadratic Equations A quadratic equation is an equation of the form ax2 bx c 0 Where a b and c are real numbers and a i 0 Solving by Factoring ZeroProduct Property If ab 0 then Example Solve by factoring x2 10x 2 9 4x2 l2x90 The Square Roots Method The solution set to x2 p p Z 0 is the set of all square roots of the number p that is xzi Example Solve using the Square Roots Method x2 25 x223 Solving Quadratic Equations by Completing the Square ax2bxc0 a 0 Make sure that a 1 if not divide each term by a 2 Get the constant on the righthand side of the equation 3 Take 12 of the coefficient ofx square it and add this number to both sides of the equation 4 Factor the lefthand side into a perfect square Solve for x using the Square Roots Method p A U Example Solve by completing the square x2 8x30 The Quadratic Formula The quadratic equation ax2bxc0 a 0 has the solutions The quantity b2 4ac denoted D is called the discriminant The equation has m distinct real roots if D gt 0 repeated real root if D 0 m real solutions if D lt 0 Note If a lt 0 multiply both sides of the equation by a l to make use of the quadratic formula easier The Halfcoef cient Quadratic Formula If in the quadratic equation ax2bxe0 a 0 lbl is an even number the following formula for the roots 2 ji ae 2 a may be useful Example Use the discriminant to determine the number of real solutions of the quadratic equation 3x2 6x 3 0 Example Use the quadratic formula to solve 3x x2 l 0 3x2 2x 1020 Applications Physics Using the position equation s l6t2 u0t s0 nd the time when an object hits the ground if it is dropped from a building at a height of 320 feet Dimensions ofa Field A farmer has 380 feet of fencing to enclose two adjacent elds Find the dimensions that would enclose an area of 4000 square feet Constructing a Box An open box is formed by cutting 15 inch squares from each corner of a rectangular piece of metal Whose length is twice its Width and bending up the edges Ifthe box is to have a volume of 21 cubic inches What dimensions should the piece of metal have Complex Numbers The number i is a solution of the equation x2 1 that is i2 1 Another solution ofthis equation is i The imaginary number i is called the 39 039 y 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 Eguali of Complex Numbers a ib e id ifand only if Sum of Complex Numbers a ib c id 2 Product of Complex Numbers a ibe id 2 Note The product of complex numbers is easy to nd by using FOIL and the fact that i2 1 Complex Conjugates The complex conjugate ofa number 2 a bi is 7a n Example Write expressions in the standard form zEabia bi 2 2 a bi a bi z a bia bi 222a2 b2 Quotient of Complex Numbers When evaluating the quotient of two complex numbers abz m the conjugate of the denominator and simplify multiply both the numerator and denominator by Example Write in the standard form 1 i 1i Powers of 139 1391 i2 i3 i4 i5 Example Evaluate The Principle Square Root of a Negative Number IfN gt 0 then J N 14 is called the principle square root of N Note 1 139 is the principle square root of 1 since 1 Ni 11 Another square root of l is z39 Caution When working with square roots of negative numbers rst rewrite the root according to the de nition above then apply any other rule for radicals 4343 W Example Find the real numbers a and b so that the equation is true a 62bi 4 5139 Example Write in standard form 3J3 03 413916 Example Perform the operations and write the result in standard form 02 30 750 J3sz 2706 20 mf 7iX 7 i 4 5 1 5 Quadratic Equations with a Negative Discriminant The quadratic equation ax2bxc0 a 0 has solutions b ixlbz 4ac a If the discriminant 2 4ac lt 0 then the quadratic equation has two complex conjugate roots x Example Use the quadratic formula to solve the equation in the complex number system x2 6x 10 2 0
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