9-5 Completing the Square

Quick Review

You can solve any quadratic equation by writing it in the form x 2 + b x = c , x squared , plus , b x equals c comma  completing the square, and finding the square roots of each side of the equation.

Example

What are the solutions of x 2 + 8 x = 513 ? x squared , plus 8 x equals 513 question mark

x 2 + 8 x + 16 = 513 + 16 Add ( 8 2 ) 2 , or 16 , to each side . ( x + 4 ) 2 = 529 Write  x 2 + 8 x + 16   as a square . x + 4 = ± 529 Find the square roots . x + 4 = ± 23 Simplify . x + 4 = 23 or x + 4 = − 23 Write as two equations . x = 19 or x = − 27 Solve for  x . table with 2 rows and 1 column , row1 column 1 , table with 4 rows and 3 columns , row1 column 1 , x squared , plus 8 x plus 16 , column 2 equals 513 plus 16 , column 3 cap add . open , 8 halves , close squared . comma or 16 comma . toeachside . . , row2 column 1 , open , x plus 4 , close squared , column 2 equals 529 , column 3 cap write . x squared , plus 8 x plus 16 . asasquare . . , row3 column 1 , x plus 4 , column 2 equals plus minus , square root of 529 , column 3 cap findthesquareroots . . , row4 column 1 , x plus 4 , column 2 equals plus minus 23 , column 3 cap simplify , . , end table , row2 column 1 , table with 2 rows and 6 columns , row1 column 1 , x plus 4 , column 2 equals 23 , column 3 or , column 4 x plus 4 , column 5 equals negative 23 , column 6 cap writeastwoequations . . , row2 column 1 , x , column 2 equals 19 , column 3 or , column 4 x , column 5 equals negative 27 , column 6 cap solvefor . x . , end table , end table

Exercises

Solve each equation by completing the square. If necessary, round to the nearest hundredth.

30. x 2 + 6 x − 5 = 0 x squared , plus 6 . x minus 5 equals 0
31. x 2 = 3 x − 1 x squared , equals 3 . x minus 1
32. 2 x 2 + 7 x = − 6 2 x squared , plus 7 x equals negative 6
33. x 2 + 10 x = − 8 x squared , plus 10 . x equals negative 8
34. 4 x 2 − 8 x = 24 4 x squared , minus 8 x equals 24
35. x 2 − 14 x + 16 = 0 x squared , minus 14 . x plus 16 equals 0
36. Construction You are planning a rectangular patio with length that is 7 ft less than three times its width. The area of the patio is 120  ft 2 . 120 , ft squared , .  What are the dimensions of the patio?
37. Design You are designing a rectangular birthday card for a friend. You want the card's length to be 1 in. more than twice the card's width. The area of the card is 88  in. 2 . 88 , in. squared , .  What are the dimensions of the card?

9-6 The Quadratic Formula and the Discriminant

Quick Review

You can solve the quadratic equation a x 2 + b x + c = 0 , eh x squared , plus b x plus c equals 0 comma  where a ≠ 0 , eh not equal to 0 comma  by using the quadratic formula x = − b ± b 2 − 4 a c 2 a . x equals . fraction negative b plus minus . square root of b squared , minus 4 eh c end root , over 2 eh end fraction . .  The discriminant is b 2 − 4 a c . b squared , minus 4 . eh c .

The discriminant tells you how many real-number solutions the equation has.

Example

How many real-number solutions does the equation x 2 + 3 = 2 x x squared , plus 3 equals 2 x  have?

x 2 − 2 x + 3 = 0 Write in standard form . b 2 − 4 a c = ( − 2 ) 2 − 4 ( 1 ) ( 3 ) Evaluate discriminant .   = − 8 Simplify . table with 3 rows and 3 columns , row1 column 1 , x squared , minus 2 x plus 3 , column 2 equals 0 , column 3 cap writeinstandardform . . , row2 column 1 , b squared , minus 4 eh c , column 2 equals . open , negative 2 , close squared . minus 4 , open 1 close . open 3 close , column 3 cap evaluatediscriminant . . , row3 column 1 , , column 2 equals negative 8 , column 3 cap simplify , . , end table

Because the discriminant is negative, the equation has no real-number solutions.

Exercises

Find the number of real-number solutions of each equation.

38. x 2 + 7 x − 10 = 3 x squared , plus 7 . x minus 10 equals 3
39. 3 x 2 − 2 = 5 x 3 x squared , minus 2 equals 5 x

Solve each equation using the quadratic formula. Round to the nearest hundredth.

40. 4 x 2 + 3 x − 8 = 0 4 x squared , plus 3 x minus 8 equals 0
41. 2 x 2 − 3 x = 20 2 x squared , minus 3 x equals 20
42. − x 2 + 8 x + 4 = 5 negative , x squared , plus 8 . x plus 4 equals 5
43. 64 x 2 + 12 x − 1 = 0 64 x squared , plus 12 x minus 1 equals 0

Solve each equation using any method. Explain why you chose the method you used.

44. 5 x 2 − 10 = x 2 + 90 5 x squared , minus 10 equals . x squared , plus 90
45. x 2 − 6 x + 9 = 0 x squared , minus 6 . x plus 9 equals 0
46. Vertical Motion A ball is thrown into the air. The height h, in feet, of the ball can be modeled by the equation h = − 16 t 2 + 20 t + 6 , h equals negative , 16 t squared , plus 20 t plus 6 comma  where t is the time, in seconds, the ball is in the air. When will the ball hit the ground?

Table of Contents