Dynamic Activity

Completing the Square

Problem 2 Determining Dimensions

Architecture While designing a house, an architect used windows like the one shown here. What are the dimensions of the window if it has 2766 square inches of glass?

• Step 1 Find the area of the window.

  The area of the rectangular part is ( 2 x ) ( x ) = 2 x 2  in. 2 . open 2 x close open x close equals , 2 x squared . in. squared , .

  The area of the semicircular part is 1 2 π r 2 = 1 2 π ( x 2 ) 2 = 1 2 π x 2 4 = π 8 x 2 in . 2 . 1 half , pi , r squared , equals , 1 half , pi . open , x over 2 , close squared . equals , 1 half , pi , fraction x squared , over 4 end fraction , equals , pi over 8 , x squared , in , . squared , .

• So, the total amount of glass used is 2 x 2 + π 8 x 2 = 2766  in . 2 . 2 , x squared , plus , pi over 8 , x squared , equals , 2766 , in , . squared , .

• Step 2 Solve for x.

  ( 2 + π 8 ) x 2 = 2766 Write the equation in  a x 2 = c  form . x 2 = 2766 2 + π 8 Isolate  x 2 . x ≈ ± 34 Find square roots . Use a calculator . table with 3 rows and 3 columns , row1 column 1 , open . 2 plus , pi over 8 . close . x squared , column 2 equals , 2766 , column 3 cap writetheequationin . eh , x squared , equals c , form , . , row2 column 1 , x squared , column 2 equals . fraction 2766 , over 2 plus , pi over 8 end fraction , column 3 cap isolate . x squared , . , row3 column 1 , x , column 2 almost equal to plus minus 34 , column 3 cap findsquareroots . .cap useacalculator . . , end table

Length cannot be negative. So the rectangular portion of the window is 34 in. wide by 68 in. long. The semicircular top has a radius of 17 in.

✓ Got It?

2. The lengths of the sides of a rectangular window have the ratio 1.6 to 1. The area of the window is 2822 . 4  in. 2 . 2822 , . 4 , in. squared , .  What are the window dimensions?

Problem 3 Solving a Perfect Square Trinomial Equation

What is the solution of x 2 + 4 x + 4 = 25 ? bold italic x squared , plus 4 bold italic x plus 4 equals 25 question mark

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3. What is the solution of x 2 − 14 x + 49 = 25 ? x squared , minus 14 x plus 49 equals 25 question mark