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# Math 111 Chapter 1 Sections 1 & 2 Reviews

 Mixtures : (1st % × Amt) + (2nd % × Amt) = Final % × Amt How many gallons of cream containing 25% butter fat and milk containing 3.5% butter fat must be mixed to obtain 50 gallons of cream containing 12.5% butter fat? Let x = # gal cream Let 50- x = # gal milk 0.25x = # gal of butter fat in cream 0.35(50 - x) = # gal of butter fat in milk 0.125(50) = # gal of butterfat in mixture 0.25x + 0.035(50 - x) = 0.125(50) x = 20.93 (rounded) Your 8 quart radiator is full with a 30% antifreeze mixture. How much pure antifreeze should be added to get a 50% antifreeze mixture. In the chemistry lab you have 20 oz of a 10% acid solution. How much 60% acid solution should you add to get a 35% acid solution? Work Problems: (part done by A ) + (part done by B) = 1 whole job Ron, Mike, and Tim are going to paint a house together. Ron can paint one side of the house in 4 hours. To paint an equal area, Mike takes only 3 hours and Tim 2 hours. If the men work together, how long will it take them to paint one side of the house? Let t be time needed to paint the side. (1/4)t + (1/3)t + (1/2)t = 1 Section 1.2 Quadratic Equations and Their Applications Solve by factoring Solve by taking the square root Solve by completing the square Solve by using the Quadratic Formula The discriminant of a Quadratic Equations Applications A quadratic equation is an equation that can be written in the following standard form: ax² + bx + c = 0, where a,b,c are real numbers and a ≠ 0 The Zero Product Principle If A and B are algebraic expressions such that AB = 0, then A = 0 or B = 0 1.2: Factoring and using Zero Product Principle Solve by factoring... x2 + x = 12Solve by factoring... 2x2 - 5x = 12 1.2: Square Root Procedure If x2 = c, then Solve by using the square root method. 2x2 = 48 Solve by using the square root method. (x + 2)2 = 36 Geometrically Completing the Square 1.2: Completing the Square Solve by completing the square 1.2: Completing the Square Solve by "completing the square"x2 - 2x- 5 = 10 2 x2 + 4x - 4= 0 The Quadratic Formula The solutions of the equation are : a ≠ 0 Solve by using the quadratic equation : 12x2 - x - 6 = 0 x2 = 2x - 2 1.2: Discriminant and solutions to the quadratic equation The equation ax2 + bx + c = 0 , with real coefficients and a≠0, has as its discriminant b2 - 4ac If b2 - 4ac > 0 then two distinct real solutions. If b2 - 4ac = 0 then one real solution. If b2 - 4ac < 0 then two distinct nonreal complex solutions. The solutions are conjugates of each other Determine discriminant and state the number of solutionsa) 4x2 - 5x + 3 = 0 b) x2 + 2x - 15 = 0 c) 4x2 + 12x + 9 = 0 Now graphically 1.2: ApplicationsThe Pythagorean Theorem If a and b denote the lengths of the legs of a right triangle and c the length of the hypotenuse, then a2 + b2 = c2 Garfield's Proof of the Pythagorean Theorem