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Sample Test: Quadratic Equations I

Module by: Kenny M. Felder. E-mail the author

Summary: A first sample test on quadratics equations.

Exercise 1

Multiply:

  • a. ( x − 3 2 ) 2 ( x − 3 2 ) 2 size 12{ \( x - { {3} over {2} } \) rSup { size 8{2} } } {}
  • b. ( x − 3 ) 2 ( x − 3 ) 2 size 12{ \( x - sqrt {3} \) rSup { size 8{2} } } {}
  • c. ( x − 7 ) ( x + 7 ) ( x − 7 ) ( x + 7 ) size 12{ \( x - 7 \) \( x+7 \) } {}
  • d. ( x − 2 ) ( x 2 − 4x + 4 ) ( x − 2 ) ( x 2 − 4x + 4 ) size 12{ \( x - 2 \) \( x rSup { size 8{2} } - 4x+4 \) } {}
  • e. ( x + 3 ) ( 2x − 5 ) ( x + 3 ) ( 2x − 5 ) size 12{ \( x+3 \) \( 2x - 5 \) } {}
  • f. Check your answer to part (e) by substituting in the number 1 for xx size 12{x} {} into both the original expression, and your resultant expression. Do they come out the same? (No credit here for just saying “yes”—I have to be able to see your work!)

Exercise 2

Here is a formula you probably never saw, but it is true: for any xx size 12{x} {} and aa size 12{a} {}, (x+a)4=x4+4x3a+6x2a2+4xa3+a4(x+a)4=x4+4x3a+6x2a2+4xa3+a4 size 12{ \( x+a \) rSup { size 8{4} } =x rSup { size 8{4} } +4x rSup { size 8{3} } a+6x rSup { size 8{2} } a rSup { size 8{2} } +4 ital "xa" rSup { size 8{3} } +a rSup { size 8{4} } } {}. Use that formula to expand the following.

  • a. ( x + 2 ) 4 = ( x + 2 ) 4 = size 12{ \( x+2 \) rSup { size 8{4} } ={}} {}
  • b. ( x − 1 ) 4 = ( x − 1 ) 4 = size 12{ \( x - 1 \) rSup { size 8{4} } ={}} {}

Exercise 3

Factor:

  • a. x 2 − 36 x 2 − 36 size 12{x rSup { size 8{2} } - "36"} {}
  • b. 2x 2 y − 72 y 2x 2 y − 72 y size 12{2x rSup { size 8{2} } y - "72"y} {}
  • c. Check your answer to part (b) by multiplying back. (*I have to see your work!)
  • d. x 3 − 6x 2 + 9x x 3 − 6x 2 + 9x size 12{x rSup { size 8{3} } - 6x rSup { size 8{2} } +9x} {}
  • e. 3x 2 − 27 x + 24 3x 2 − 27 x + 24 size 12{3x rSup { size 8{2} } - "27"x+"24"} {}
  • f. x 2 + 5x + 5 x 2 + 5x + 5 size 12{x rSup { size 8{2} } +5x+5} {}
  • g. 2x 2 + 5x + 2 2x 2 + 5x + 2 size 12{2x rSup { size 8{2} } +5x+2} {}

Exercise 4

Geoff has a rectangular yard which is 55' by 75'. He is designing his yard as a big grassy rectangle, surrounded by a border of mulch and bushes. The border will be the same width all the way around. The area of his entire yard is 4125 square feet. The grassy area will have a smaller area, of course—Geoff needs it to come out exactly 3264 square feet. How wide is the mulch border?

Exercise 5

Standing outside the school, David throws a ball up into the air. The ball leaves David’s hand 4' above the ground, traveling at 30 feet/sec. Raven is looking out the window 10' above ground, bored by her class as usual, and sees the ball go by. How much time elapsed between when David threw the ball, and when Raven saw it go by? To solve this problem, use the equation h(t)=h0vot−16t2h(t)=h0vot−16t2 size 12{h \( t \) =h rSub { size 8{0} } v rSub { size 8{o} } t - "16"t rSup { size 8{2} } } {}.

Exercise 6

Solve by factoring: 2x2−11x−30=02x2−11x−30=0 size 12{2x rSup { size 8{2} } - "11"x - "30"=0} {}

Exercise 7

Solve by completing the square: 2x2+6x+4=02x2+6x+4=0 size 12{2x rSup { size 8{2} } +6x+4=0} {}

Exercise 8

Solve by using the quadratic formula: −x2+2x+1=0−x2+2x+1=0 size 12{ - x rSup { size 8{2} } +2x+1=0} {}

Exercise 9

Solve. (*No credit unless I see your work!) ax 2 + bx + c = 0 ax 2 + bx + c = 0 size 12{ ital "ax" rSup { size 8{2} } + ital "bx"+c=0} {}

Solve any way you want to:

Exercise 10

2x 2 + 4x + 10 = 0 2x 2 + 4x + 10 = 0 size 12{2x rSup { size 8{2} } +4x+"10"=0} {}

Exercise 11

( 1 2 ) x 2 − x + 2 1 2 = 0 ( 1 2 ) x 2 − x + 2 1 2 = 0 size 12{ \( { {1} over {2} } \) x rSup { size 8{2} } - x+2 { {1} over {2} } =0} {}

Exercise 12

x 3 = x x 3 = x size 12{x rSup { size 8{3} } =x} {}

Exercise 13

Consider the equation 3x2−bx+2=03x2−bx+2=0 size 12{3x rSup { size 8{2} } - ital "bx"+2=0} {}, where bb size 12{b} {} is some constant. For what values of bb size 12{b} {} will this equation have…

  • a. No real answers:
  • b. Exactly one answer:
  • c. Two real answers:
  • d. Can you find a value of b for which this equation will have two rational answers—that is, answers that can be expressed with no square root? (Unlike (a)-(c), I’m not asking for all such solutions, just one.)

Extra Credit:

(5 points) Make up a word problem involving throwing a ball up into the air. The problem should have one negative answer and one positive answer. Give your problem in words—then show the equation that represents your problem—then solve the equation—then answer the original problem in words.

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