10.Solving Quadratic Equations

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Passport to advanced mathematics Passport to advanced mathematics Part I solving quadratic equations Part I solving quadratic equations 1

1.. WWhhiicch h oof f tthhe e ffoolllloowwiinng g ccoouulld d bbe e tthhe e ffaaccttoorreed d ffoorrm m oof f tthhee polynomial shown below?

polynomial shown below? x x 22₊₃₊₃_x_x₋₁₈₋₁₈ A. A.  x x −3!−3! x x +"!+"! #. #.  x x +3!+3! x x −"!−"! $. $.  x x −3!−3! x x −"!−"! %. %.  x x +3!+3! x x +"!+"! $

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2

polynomial shown below? x x 22+1"+1" x x +"(+"( A. A.  x x +"!+"! x x +1(!+1(! #. #.  x x +"!+"! x x −1(!−1(! $. $.  x x −"!−"! x x +1(!+1(! %. %.  x x −"!−"! x x −1(!−1(! $

3

polynomial shown below? x x 22₊₁₂₊₁₂_x_x₊₂₎₊₂₎ A. A.  x x −3! −3!  x x +*!+*! #. #.  x x +3! +3!  x x −*!−*! $. $.  x x −3! −3!  x x −*!−*! %. %.  x x +3! +3!  x x +*!+*! $orr

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.. WWhhiicch h oof f tthhe e ffoolllloowwiinng g ccoouulld d bbe e tthhe e ffaaccttoorreed d ffoorrm m oof f tthhee polynomial shown below?

polynomial shown below? x x 22₋₋_x_x_−"_−" A. A.  x x −3! −3!  x x +2!+2! #. #.  x x +3! +3!  x x +2!+2! $. $.  x x −3! −3!  x x −2!−2! %. %.  x x +3! +3!  x x −2!−2! $

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,.. WWhhiicch h oof f tthhe e ffoolllloowwiinng g ccoouulld d bbe e tthhe e ffaaccttoorreed d ffoorrm m oof f tthhee polynomial shown below?

polynomial shown below? x x 22+3+3 x x −28−28 A. A.  x x +)! +)!  x x −!−! #. #.  x x −)! −)!  x x −!−!

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$.

$.  x x +)! +)!  x x +!+! %.

%.  x x −)! −)!  x x +!+! $

"

".. 11((((−1−12211k k 22_-(_-(

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A A.. 00-- 100100 121 121 # #.. 00--−− 100100₁₂₁₁₂₁ and 0- and 0- 100100₁₂₁₁₂₁ $ $.. 00-- 1010 1 1 11 % %.. 00--−− ₁_{1 1}1010₁ and 0- and 0- 11 0₁₁₁₁0 $

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).. 22 x x −3!−3! x x +!-(+!-( et

et x x --aa and and x x --bb be the solutions to the euation abo/e. What is be the solutions to the euation abo/e. What is the /alue of −

the /alue of −aa−−bb?? $orr

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8

8.. ((--22 y y −1!8−−1!8− y y !! et

et y y --uu and and y y --dd be the solutions to the euation abo/e. What is the be the solutions to the euation abo/e. What is the /alue of

/alue of u udd?? $orrect

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*

*.. 22 x x +,!−+,!−mx mx +*!-(+*!-( 4n

4n the the eueuatiation on aboabo/e5/e5 mm is is a a coconsnstatantnt. . 4f 4f ththe e eeuauatition on hahas s ththee solutions solutions x x -−-− 55 2 2 and and x x - -3 3 2

2 5 what is the /alue of 5 what is the /alue of mm??

$orrect

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1(. 1(. x x ++ 11 33 2 2 !! x x −− 1 1 33 2 2 !-(!-(

6ow many distinct real solutions does the

6ow many distinct real solutions does the euation abo/e ha/e?euation abo/e ha/e? A. A. (( #. #. 11 $. $. 22 %. %.  $

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11. 11. 22 3 3 x x 2 2₋₋ 11 2 2 x x −− 3 3 4 4 - ( - (

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et 7-w and 7- be the solutions to the euation abo/e. What is et 7-w and 7- be the solutions to the euation abo/e. What is the /alue of w+? the /alue of w+? A A.. −− 33 2 2 # #.. 99 33 4 4 $. $. (( %. %. 33 4 4 $

12.2 12.2 x x 22₋₋ 1111 2 2 x x −− 3 3 2 2 - ( - (

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A A.. 77- - −− 11 4 4 and and x x - 2- 2 # #.. 77- - −− 11 4 4 and and x x - 3- 3 $. $. 7- 7- 2 2 aandnd x x - 3- 3 % %.. 77- - −− 11 4 4 5 7- 25 and 5 7- 25 and x x - 3- 3 $

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13.

13.t t +1,!+1,!22_+,-(_+,-(

1.

1. x x 22−13−13 x x +3(-(+3(-(

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A. A. 7-7-−1−1, a, and nd 7-7-22 #. #. 7-7-−1( −1( anand 7d 7-:-:33 $. $. 7 - 7 - 3 a3 and nd 7-7-1(1( %. %. 7-7-:2 :2 anand 7d 7-1-1,, $

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1,.2

1,.2 x x 22+,+, x x −−k k -(-(

4n the euation abo/e5

4n the euation abo/e5 k k is a constant. ;or what /alue of is a constant. ;or what /alue of k k does the does the euation ha/e e7actly one distinct real solution?

euation ha/e e7actly one distinct real solution? A.

A. −−2525

8 8

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#. #. −−55 4 4 $. $. 55 4 4 %. %. 2525 8 8 $

1".

1".t t −1(!−1(!22_−1"-(_−1"-(

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A. A. t-t- a and nd t-t-3"3" #. #. t-t-" a" and nd t-t-11 $. $. t-t-1( 1( anand td t-−1-−1"" %. %. t-t-1( 1( anand td t-1-1"" $

1).

1). x x −1-2−−1-2− x x !!22

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A A.. 77--(( # #.. 77-- 55−−√ √ 55 2 2 and 7- and 7-5 5₊₊√ √ 55 2 2 $ $.. 77-- −−11−−√ √ 2121 2 2 and 7- and 7-− −11++√ √ 2121 2 2 %.

%. <he eu<he euation hation has no ras no real soluteal solutions.ions. $

18.− 18.− 11 2 2 t t −3!+−3!+t t 2 2_-(_-(

1*

1*.. −)−) x x −8!−8!22_-−"3_-−"3

et

et x x --k k and and x x --ll be solutions to the euation abo/e. What is the be solutions to the euation abo/e. What is the /alue of

/alue of k k ll?? $orrect

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2(.

2(.ax ax 22_+,_+,_x_x_+2-(_+2-(

4n

4n the the eueuatiation on aboabo/e5/e5 aa is is a a coconsnstatantnt. . 4f 4f ththe e eeuauatition on hahas s ththee solutions

solutions x x -−2 and-−2 and x x -−-− 11

2

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21.3

21.3nn22-2)-2)

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A A.. nn-- _√_√33 # #.. nn--33 $ $.. nn-- −−√ √ 33 and n- and n- √ √ 33 % %.. nn-- −−¿¿ 3 and n-33 and n-3 $

22

22.. WhWhicich h of of ththe e fofollllowowining g uuadadraratitic c eeuauatitionons s hahas s e7e7acactltly y ononee distinct real solution?

distinct real solution? A. A. 99 16 16 x x 2 2₋₃₋₃_x_x_−-(_−-( #. #. 99 16 16 x x 2 2₋₃₋₃_x_x_+-(_+-( $. $. 1616 25 25 x x 2 2_−2,-(_−2,-( %. %. 1616 25 25 x x 2 2_+2,-(_+2,-( $

23. 23.vv −− 55 3 3 !! 2 2_-*_-*

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A. A. v v -−-− 142142 3 3 and /- and /-152 152 3 3 #. #. v v -− and-− and v v -,-, $. $. v v -−2 and-−2 and v v -12-12 %. %. v v -−-− 1616 3 3 and and v v - -26 26 3 3 $

2

2.. 11−−aa!+31−!+31−aa!!22_-(_-(

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A A.. aa--11 # #.. aa-- 44 3 3 $. $. a-a-1 a1 and nd aa-- 44 3 3 %.

%. a-a-( a( and nd a-a-−− 11

3 3

$

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2,. 2,.k k k k −− 11 2 2 !-−!-− 1 1 16 16

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A A.. 00-- 11 8 8 # #.. 00-- 11 4 4 $ $.. 00--−− 11 8 8 and 0- and 0-1 1 8 8 % %.. 00--−− 11 4 4 and 0- and 0-1 1 4 4 $

2".

2".t t ++ 88

3

3 !!t t ++bb!-(!-(

4n

4n ththe e eeuauatition on ababo/o/e5e5 bb iis s a a ccoonnssttaanntt. . 44f f −− 88

3 3 and and 13 13 3 3 are are

solutions to the euation5 then what is the /alue of solutions to the euation5 then what is the /alue of bb?? A A.. −− 1313 3 3 # #.. −− 88 3 3 $. $. 88 3 3 %. %. 1313 3 3 $

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2). 2).vv ++ 11 5 5 ! ! 2 2_{− *-(}_{− *-(}

What is the sum of the solutions to the euation abo/e? What is the sum of the solutions to the euation abo/e? A A.. −− 33 5 5 # #.. −− 22 5 5 $ $.. −− 11 5 5 %. %. (( $

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28.

28. x x −−aa!!22_−3-(_−3-(

4n

4n the the eueuatiation on aboabo/e5/e5 aa is is a a coconsnstatantnt. . 4f 4f ththe e eeuauatition on hahas s ththee solutions 7-=>3 5 what is the /alue of

solutions 7-=>3 5 what is the /alue of aa?? $orrect

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2*.

2*. x x +1!+1!cc++ x x 22-(-(

44n n tthhe e eeuuaattiioon n aabboo//ee55 cc iis s a a ccoonnssttaanntt.. <he euation has solutions at −

<he euation has solutions at − 33

2

2 and −3. What is the /alue of and −3. What is the /alue of cc??

A. A. 33 2 2 #. #. 33 $. $. 99 2 2 %. %. ** $orrect

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3(.2

3(.2 x x −3!−3!22₋₋_b_b_-(_-(

4n

4n the the eueuatiation on aboabo/e5/e5 bb is is a a coconsnstatantnt. . 4f 4f ththe e eeuuatatioion n hahas s ththee solutions 7-3=>,5 what is the /alue of

solutions 7-3=>,5 what is the /alue of bb?? $orr

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31 31.. − −  1212 8 8 − − bb!- −!- − bb 2 2 4f s and

b-4f b-s and b-t t are the solutions to the euation abo/e5 what is the are the solutions to the euation abo/e5 what is the /alue of /alue of st st ?? A A.. −− 33 2 2 # #.. −− 22 3 3 $. $. 22 3 3 %. %. 33 2 2 $orr

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32.3

32.3 x x 22_+_+_x_x₋₋_k_k_-(_-(

4n the euation abo/e5 k k is a constant. ;or what /alue of is a constant. ;or what /alue of k k does the does the euation ha/e no real solutions?

euation ha/e no real solutions? A A.. −− 44 3 3 #. #. 44 3 3 $ $.. −− 55 3 3 %. %. 55 3 3 $orrect

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33. 33. 11 2 2 x x 2 2₋₋ 11 6 6 x x −− 1 1 3 3 - ( - (

What are the solutions to the euation abo/e? What are the solutions to the euation abo/e? A. 7-1 and 7-− A. 7-1 and 7-− 22 3 3 #. 7 -#. 7 - 11 6 6 − −

√√

23 23 36 36 and and 7 7 - -1 1 6 6 + +

√√

23 23 36 36 $. 7-2 and 7-− $. 7-2 and 7-− 44 3 3 %. 7 -%. 7 - 11 6 6 − −

√√

− −2323 36 36 and and 7 7 - -1 1 6 6 + +

√√

− −2323 36 36 $orr

3.

3.t t +1!+1!22₊₊_c_c_-(_-(

44n n tthhe e eeuuaattiioon n aabboo//ee55 cc iis s a a ccoonnssttaanntt.. <he

<he euation euation has has solutions solutions at t-at t- 33

2

2 a and nd t-t-−− 7 7 2

2 . What is the . What is the

/alue of /alue of cc?? A A.. −− 729729 4 4 # #.. 99 121121 4 4 $ $.. 99 2525 4 4 % %.. 9 9 11 $orrect

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3,.

3,. x x 22++kx kx −1-(−1-(

4n the euation abo/e5 k k is a constant. <he euation has solutions is a constant. <he euation has solutions at ) and −2 . What is the /alue of

at ) and −2 . What is the /alue of k k ?? A A.. −−** # #.. −−,, $. $. ,, %. %. ** $orrect

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3".

3".at at 22++ 77

2

2 tt − -(− -(

<he

<he abo/e abo/e euation euation has has solutions solutions atat t t -−-−8 a8 andnd t t -1. What is the-1. What is the /alue of the constant

/alue of the constant aa?? $orr

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3).+

3).+kpkp-- p p22

4n the euation abo/e5 k k is a constant. 4f the sum of the solutions to is a constant. 4f the sum of the solutions to the euation is (5 what is the /alue of

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A. A. (( #. #. 11 $. $. 22 %. %.  $orr

38.1"

38.1" x x 22−8−8 x x −3-(−3-( et

et x x --qq and and x x --r r be solutions to the euation shown abo/e5 with be solutions to the euation shown abo/e5 with q

qr r . What is the /alue of . What is the /alue of q q−−r r ?? $orrect