Problem Solving With Quadratic Equations

For some problems, the solution can be obtained by representing the problem with a quadratic equation, and solving this equation using any one of the methods discussed in the preceding section. Whenever the given conditions require you to multiply an expression containing a variable, the result is a quadratic equation. Take a look at the following word problems that yield algebraic equations.

Problem 1: A picture that measures 4 inches by 6 inches is surrounded by a border of uniform width. If the total area of the border and the frame is twice the area of the picture, how wide is the border?Solution: 64

6+2x4+2x

Let x be the width of the border. The dimensions of the larger rectangle are 4+2x and 6+2x. Since the new area is twice the old area, you can form the following equations:(6+2x)(4+2x) = 2(6)(4)Simplifying, you get4x2+20x+24 = 484x2+20x-24 = 04(x2+5x-6) = 04(x-1)(x+6) = 0(x-1)(4x+24) = 0x = 1, -6Since x cannot be negative, the correct answer is x=1. The border is 1 inch wide all over. To check the correctness of the answer, the dimensions of the bigger rectangle are 4+2 and 6+2, or 6 and 8. Hence, its area is (6)(8) = 48 in2, which is twice the area of the picture.

Problem 2: The sum of two integers is 12, and their product is 32. What is the larger number?Solution:Let the first number be x. Since the sum of the two numbers is 12, the second number may be represented by 12-x. Since the product is 32, you get the equation x(12-x) = 3212x-x2-32 = 0x2-12x+32 = 0(x-4)(x-8) = 0x=4, x=8If x=4, then 12-x=8. If x=8, then 12-x=4. This shows that the two integers are 4 and 8, and the larger number is 8.

Problem 3: If they worked together, John and Vince can finish their project in Biology in two days. If they work individually, it will take John three days longer than Vince to complete the task. How long will it take each to do the job alone?SolutionThe information from the problem can be summarized in the following table:Number of days to Do the JobPart of the Job Done in One DayJohn x+3Vince xTogether 2 ½

The part of the task done by the two if they work together is the combined part of the work that they can do individually, so you get the equationGet the equivalent equationx2-x-6 = 0 = (x-3)(x+2)Thus, either x=3 or x=-2. Since x represents a number of days, it cannot be negative, so you discard -2. This gives you x=3 the number of days it takes for Vince to do the job by himselfx+3=6 the number of days John needs to do the job on his own

Exercises:Solve the following word problems.1. The height of a triangle is 3 inches less than its base. If the area of the triangle is 27 in2, find the base of the triangle.2. Jason is two years older that Aga. Three years ago, the product of their ages was 48. How old is Jason now?3. The sum of the reciprocals of two integers is . If the larger number exceeds the smaller ¾number by 2, what is the larger integer?4. A piece of string 24 inches long is to be cut into two pieces such that the product of the lengths of the two pieces is 128. How long is the shorter piece?5. An open box is to be made out of a piece of cardboard 10 in by 15 in by cutting off equal squares from the four corners and turning up the sides. Find the length of the sides of each square if the lateral surface area of the box is 68 in2?