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Chapter 11 Quadratic Equations

11.1 Review of Solving Equation by Factoring 11.2 The Square Root Property and Completing the Square 11.3 The Quadratic Formula Putting It All Together 11.4Equations in Quadratic Form 11.5Formulas and Applications 11 Quadratic Equations

Formulas and Applications 11.5 Sometimes, solving a formula for a variable involves using one of the techniques we’ve learned for solving a quadratic equation or for solving an equation containing a radical. Solve a Formula for a Variable Solve. Example 1 Solution The goal is to get me on a side by itself. Square both sides. Multiply both sides by m. Divide both sides by v 2.

Example 2 Solution

Solve an Applied Problem Involving Volume A rectangular piece of cardboard is 7 in. longer than it is wide. A square piece that measures 2 in. on each side is cut from each corner, then the sides are turned up to make an uncovered box with volume 396 in 3. Find the length and width of the original piece of cardboard. Example 3 Solution Step 1: Read the problem carefully. Draw a picture. Step 2: Choose a variable to represent the unknown, and define the other unknown in terms of this variable. Let x = the width of the cardboard x + 7 = the length of the cardboard Step 3: Translate the information that appears in English into an algebraic equation. The volume of a box is (length)(width)(height). We will use the formula (length)(width)(height) = 396. Original Cardboard Box x + 7 x + 3

Original Cardboard Box x + 7 The figure on the left shows the original piece of cardboard with the sides labeled. The figure on the right illustrates how to label the box when the squares are cut out of the corners. When the sides are folded along the dotted lines, we must label the length, width, and height of the box. = x + 7– 2 = x + 3 = x – 2 = x -4 Equation: 396 = (x+3)(x-4)(2) x + 3

Step 4: Solve the equation. 396 = (x+3)(x – 4)(2) 198 = (x+3)(x – 4) 198 = x 2 – x – 12 0 = x 2 – x – = (x – 15)(x+14) x – 15 =0 or x +14 = 0 x = 15 or x = -14 Step 5: Check the answer and interpret the solution as it relates to the problem. Because x represents the width, it cannot be negative. Therefore, the width of the original piece of cardboard is 15 in. The length of the cardboard is x + 7, so = 22 in. Width of cardboard =15 in. Length of cardboard = 22 in. Divide both sides by 2. Multiply. Write in standard form. Factor. Set each factor equal to zero. Solve.

Solve an Applied Problem Involving Area Example 4 A rectangular pond is 20 ft long and 12 ft wide. The pond is bordered by a strip of grass of uniform (the same) width. The area of the grass is 320 ft2. How wide is the border of grass around the pond? Solution Step 1: Read the problem carefully. Draw a picture. Step 2: Choose a variable to represent the unknown, and define the other unknowns in terms of this variable.

Step 3: Translate from English into an algebraic equation. We know that the area of the grass border is 320 ft 2. We can calculate the area of the pond since we know its length and width. The pond plus grass border forms a large rectangle of length x and width x. The equation will come from the following relationship: Step 4: Solve the equation. Step 5: Check the answer and interpret the solution as it relates to the problem. x represents the width of the strip of grass, so x cannot equal 20. The width of the strip of grass is 4 ft. Check: Substitute x 4 into the equation written in step 3.

Solve an Applied Problem Using a Quadratic Equation Example 5 The total tourism-related output in the United Stated from 2000 to 2004 can be Modeled by y = 16.4x 2 – 50.6x Where x is the number of years since 2000 and y is the total tourism-related output In billions of dollars. ( a)According to the model, how much money was generated in 2002 due to tourism-related output? b) In what year was the total tourism-related output about $955 billion? Solution a) Since x is the number of years after 2000, the year 2002 corresponds to x = 2. y = 16.4x 2 – 50.6x y = 16.4(2) 2 – 50.6(2) y = The total tourism-related output in 2002 was approximately $860.4 billion.

b) Since y represents the total tourism-related output (in billions), substitute 955 for y and solve for x. y = 16.4x 2 – 50.6x = 16.4x 2 – 50.6x = 16.4x 2 – 50.6x – 59 Use the quadratic formula to solve for x. a = 16.4 b = – 50.6 c = – 59 The negative value of x does not make sense in the context of the problem. Use, which corresponds to the year The total tourism-related output was About $955 billion in 2004.