Exercise 72 Page 222

We are given a rectangular field with area A=x^2-120x+3500. We are also told that the width of the rectangle is x-50 feet. Using the fact that the area of a rectangle is the product of its dimensions, we can establish the following relation. A &= Width * Length x^2-120x+3500 &= (x-50)* Length To find the length, we have to factor x^2-120x+3500. Note that this is a quadratic expression. Since this expression does not have a common factor, we will try to rewrite the linear term as the sum of two terms with coefficients that are factors of ac and have a sum of b.

x^2-120x+3500 ⇕ 1x^2+( - 120)x+ 3500 [0.8em] a c = 1( 3500) = 3500 b = - 120 Notice that we do not need to make the list of the factors of 3500. Since the expression for the area contains the factor (x-50), then one of the factors is - 50. Therefore, the other factor is - 70. - 50(- 70) = 3500 - 50 + (- 70) = - 120 We can now write the factored form of the expression for the area. A=(x+(- 50))(x+(- 70)) ⇕ A=(x-50)(x-70) The above implies the length of the rectangle is (x-70) feet.