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# BASIC COLLEGE ALGEBRA MAC 1105

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This 15 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1105 at University of Florida taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/207049/mac-1105-university-of-florida in Calculus and Pre Calculus at University of Florida.

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Date Created: 09/19/15

L6 Quadratic Equations and the Quadratic Formula Applications 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 O Solving by Factoring ZeroProduct Property If ab 0 then Example Solve by factoring x2 10x 2 9 4x2 12x9O The Square Roots Method The solution set to x2 p p Z O is the set of all square roots of the number 9 that is x id Example Solve by using the Square Roots Method x2225 x223 Solving Quadratic Equations by Completing the Square Jgt t ax2bxcO 61720 Make sure that a 1 if not divide each term by 01 Get the constant on the righthand side of the equation Take 12 of the coefficient of x square it and add this number to both sides of the equation Factor the lefthand side into a perfect square Solve for x by using the Square Roots Method Example Solve by completing the square x2 8x 3 0 Example Factor by using completing the square x24x 3 The Quadratic Formula The quadratic equation ax2 bx c O a 72 O has the solutions roots x Jain2 4ac 2a The quantity 2 4ac denoted D is called the discriminant The equation has two distinct real roots if D gt 0 repeated real root if D 0 real solutions if D lt 0 Note If a lt O multiply both sides of the equation by a l to make use of the quadratic formula easier The Halfcoefficient Quadratic Formula If in the quadratic equation ax2bxcO a 0 b is an even number the following formula for the roots 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 O 3x2 2x 1020 The Quadratic Formula and Factoring Theorem If x1 and x2 are the roots of the quadratic equation 0m2 bx c 0 then the quadratic trinomial can be factored as follows 00c2 bxc ax x1x x2 Example Factor by using the quadratic formula 4c25c i 2 Applications Example Find two consecutive positive integers such that the sum of their squares is 145 Geometm Finding the Dimensions The height of a triangular sign is equal to its base The area of the sign is 20 square feet Find the base and the height of the sign Physics Using the position equation s 16t2 Uot so find the time when an object hits the ground if it is dropped from a building at a height of 320 feet Dimensions of a Field A farmer has 380 feet of fencing to enclose two adjacent fields 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 comer of a rectangular piece of metal whose length is twice its width and bending up the edges If the box is to have a volume of 21 cubic inches what dimensions should the piece of metal have 60 L6 Quadratic Equations and the Quadratic Formula Applications 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 x i5 Example Solve by 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 by using the Square Roots Method p A U Example Solve by completing the square x2 8x30 Example Factor by using completing the square x24x 3 The Quadratic Formula The quadratic equation ax2bxc0 a 0 has the solutions roots x bilb2 4ae 2a 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 The Quadratic Formula and Factoring Theorem If x1 and x2 are the roots of the quadratic equation ax2 bx 0 2 0 then the quadratic trinomial can be factored as follows ax2 bxc ax x1x x2 Example Factor by using the quadratic formula 4x2 5x 2 Applications Example Find two consecutive positive integers such that the sum of their squares is 145 Geomet Finding the Dimensions The height of a triangular sign is equal to its base The area of the sign is 20 square feet Find the base and the height of the sign 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

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