U 3 v applications there are many problems that can

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College Algebra
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Chapter 1 / Exercise 41
College Algebra
Larson
Expert Verified
U 3 V Applications There are many problems that can be solved by equations like those we have just discussed. 366 Chapter 5 Factoring 5-46 E X A M P L E 7 Area of a garden Merida’s garden has a rectangular shape with a length that is 1 foot longer than twice the width. If the area of the garden is 55 square feet, then what are the dimensions of the garden? Solution If x represents the width of the garden, then 2 x 1 represents the length. See Fig. 5.1. Because the area of a rectangle is the length times the width, we can write the equation x (2 x 1) 55. We must have zero on the right-hand side of the equation to use the zero factor property. So we rewrite the equation and then factor: 2 x 2 x 55 0 (2 x 11)( x 5) 0 Factor. 2 x 11 0 or x 5 0 Zero factor property x 1 2 1 or x 5 The width is certainly not 1 2 1 . So we use x 5 to get the length: 2 x 1 2(5) 1 11 We check by multiplying 11 feet and 5 feet to get the area of 55 square feet. So the width is 5 ft, and the length is 11 ft. Now do Exercises 65–66 x ft 2 x 1 ft Figure 5.1 U Helpful Hint V To prove the Pythagorean theorem start with two identical squares with sides of length a b , and partition them as shown. There are eight identical triangles in the diagram. Erasing four of them from each original square will leave smaller squares with areas a 2 , b 2 , and c 2 . Since the original squares had equal areas, the remaining areas must be equal. So a 2 b 2 c 2 . b a c c c c c 2 a b b a a b b 2 b b b c c b a a a a a 2 Figure 5.2 c b a Hypotenuse Legs
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College Algebra
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Chapter 1 / Exercise 41
College Algebra
Larson
Expert Verified
5-47 5.6 Solving Quadratic Equations by Factoring 367 The hypotenuse is the longest side of a right triangle. So if the lengths of the sides of a right triangle are 5 meters, 12 meters, and 13 meters, then the length of the hypotenuse is 13 meters, and 5 2 12 2 13 2 . CAUTION E X A M P L E 8 Using the Pythagorean theorem The length of a rectangle is 1 meter longer than the width, and the diagonal measures 5 meters. What are the length and width? Solution If x represents the width of the rectangle, then x 1 represents the length. Because the two sides are the legs of a right triangle, we can use the Pythagorean theorem to get a relation- ship between the length, width, and diagonal. See Fig. 5.3. x 2 ( x 1) 2 5 2 Pythagorean theorem x 2 x 2 2 x 1 25 Simplify. 2 x 2 2 x 24 0 x 2 x 12 0 Divide each side by 2. ( x 3)( x 4) 0 x 3 0 or x 4 0 Zero factor property x 3 or x 4 The length cannot be negative. x 1 4 To check this answer, we compute 3 2 4 2 5 2 , or 9 16 25. So the rectangle is 3 meters by 4 meters. Now do Exercises 67–68 Figure 5.3 5 x x 1 Warm-Ups Fill in the blank. 1. A equation has the form ax 2 bx c 0 where a 0. 2. A equation is two equations connected with the word “or.” 3. The property says that if ab 0, then a 0 or b 0. 4. Some quadratic equations can be solved by . 5. We do not usually each side of an equation by a variable.

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