Exercise algebra.equations.quadratic.no-abc

x == sqrt 2 or x == -sqrt 2

2.

x^2+3 == 52

x^2 == 49

x == 7 or x == -7

3.

x^2-7 == 0

⇒ algebra.equations.quadratic.no-lin

x^2 == 7

x == sqrt 7 or x == -sqrt 7

4.

9*x^2-6 == 75

9*x^2 == 81

x^2 == 9

x == 3 or x == -3

5.

32-2*x^2 == 14

2*x^2 == 18

x^2 == 9

x == 3 or x == -3

6.

2*(x^2-3) == 12

x^2-3 == 6

x^2 == 9

x == 3 or x == -3

7.

1/4*x^2+12 == 16

1/4*x^2 == 4

x^2 == 16

x == 4 or x == -4

8.

(x-1)^2 == 100

x-1 == 10 or x-1 == -10

x == 11 or x-1 == -10

x == 11 or x == -9

9.

14-2*x^2 == 6

2*x^2 == 8

x^2 == 4

x == 2 or x == -2

10.

1/4*(17-x^2) == 2

17-x^2 == 8

x^2 == 9

x == 3 or x == -3

11.

(x-7)^2+3 == 11

(x-7)^2 == 8

x-7 == sqrt 8 or x-7 == -sqrt 8

x == 7+sqrt 8 or x-7 == -sqrt 8

x == 7+sqrt 8 or x == 7-sqrt 8

x == 7+2*sqrt 2 or x == 7-2*sqrt 2

12.

(6-2*x)^2 == 81

6-2*x == 9 or 6-2*x == -9

2*x == -3 or 6-2*x == -9

x == -3/2 or 6-2*x == -9

x == -3/2 or 2*x == 15

x == -3/2 or x == 15/2

13.

1/2*(x+9)^2 == 4

(x+9)^2 == 8

x+9 == sqrt 8 or x+9 == -sqrt 8

x == -9+sqrt 8 or x+9 == -sqrt 8

x == -9+sqrt 8 or x == -9-sqrt 8

x == -9+2*sqrt 2 or x == -9-2*sqrt 2

14.

(3-x^2)/10 == 2

⇒ algebra.equations.coverup.numerator

3-x^2 == 20

x^2 == -17

false

15.

5*x^2+3*x == 3*x+2

5*x^2 == 2

x^2 == 2/5

x == sqrt (2/5) or x == -sqrt (2/5)

x == 1/5*sqrt 10 or x == -1/5*sqrt 10

16.

11-(2*x+1)^2 == 5

(2*x+1)^2 == 6

2*x+1 == sqrt 6 or 2*x+1 == -sqrt 6

2*x == -1+sqrt 6 or 2*x+1 == -sqrt 6

x == (-1+sqrt 6)/2 or 2*x+1 == -sqrt 6

x == (-1+sqrt 6)/2 or 2*x == -1-sqrt 6

x == (-1+sqrt 6)/2 or x == (-1-sqrt 6)/2

x == -1/2+sqrt 6/2 or x == -1/2-sqrt 6/2

17.

(6*x-3)^2+6 == 12

(6*x-3)^2 == 6

6*x-3 == sqrt 6 or 6*x-3 == -sqrt 6

6*x == 3+sqrt 6 or 6*x-3 == -sqrt 6

x == (3+sqrt 6)/6 or 6*x-3 == -sqrt 6

x == (3+sqrt 6)/6 or 6*x == 3-sqrt 6

x == (3+sqrt 6)/6 or x == (3-sqrt 6)/6

x == 1/2+sqrt 6/6 or x == 1/2-sqrt 6/6

18.

(7+2*x)^2 == 7

7+2*x == sqrt 7 or 7+2*x == -sqrt 7

2*x == -7+sqrt 7 or 7+2*x == -sqrt 7

x == (-7+sqrt 7)/2 or 7+2*x == -sqrt 7

x == (-7+sqrt 7)/2 or 2*x == -7-sqrt 7

x == (-7+sqrt 7)/2 or x == (-7-sqrt 7)/2

x == -7/2+sqrt 7/2 or x == -7/2-sqrt 7/2

19.

4-x^2/10 == 6

x^2/10 == -2

⇒ algebra.equations.coverup.numerator

x^2 == -20

false

20.

12-(2*x+3)^2 == 6

(2*x+3)^2 == 6

2*x+3 == sqrt 6 or 2*x+3 == -sqrt 6

2*x == -3+sqrt 6 or 2*x+3 == -sqrt 6

x == (-3+sqrt 6)/2 or 2*x+3 == -sqrt 6

x == (-3+sqrt 6)/2 or 2*x == -3-sqrt 6

x == (-3+sqrt 6)/2 or x == (-3-sqrt 6)/2

x == -3/2+sqrt 6/2 or x == -3/2-sqrt 6/2

21.

x^2 == 5*x

x^2-5*x == 0

x*(x-5) == 0

x == 0 or x == 5

22.

x^2-6*x == 0

x*(x-6) == 0

x == 0 or x == 6

23.

6*x+x^2 == 0

x*(x+6) == 0

x == 0 or x == -6

24.

x*(x+4) == 0

x == 0 or x == -4

25.

x*(2*x-4) == 0

x == 0 or 2*x-4 == 0

x == 0 or 2*x == 4

x == 0 or x == 2

26.

3*x^2 == 6*x

x^2 == 2*x

x^2-2*x == 0

x*(x-2) == 0

x == 0 or x == 2

27.

3*x == 2*x^2

-2*x^2+3*x == 0

-x*(2*x-3) == 0

x == 0 or 2*x-3 == 0

x == 0 or 2*x == 3

x == 0 or x == 3/2

28.

x*(1-6*x) == 0

x == 0 or 1-6*x == 0

x == 0 or 6*x == 1

x == 0 or x == 1/6

29.

(x+5)*(x-8) == 0

x == -5 or x == 8

30.

(3*x-1)*(x+3) == 0

3*x-1 == 0 or x == -3

3*x == 1 or x == -3

x == 1/3 or x == -3

31.

x^2-2*x == 3

x^2-2*x-3 == 0

(x+1)*(x-3) == 0

x == -1 or x == 3

32.

x^2+12*x+20 == 0

(x+2)*(x+10) == 0

x == -2 or x == -10

33.

x^2-x == 30

x^2-x-30 == 0

(x+5)*(x-6) == 0

x == -5 or x == 6

34.

x*(x+2) == 8

x^2+2*x == 8

x^2+2*x-8 == 0

(x-2)*(x+4) == 0

x == 2 or x == -4

35.

x*(x-3) == 4

x^2-3*x == 4

x^2-3*x-4 == 0

(x+1)*(x-4) == 0

x == -1 or x == 4

36.

2*x+15 == x^2

-x^2+2*x+15 == 0

(x+3)*(x-5) == 0

x == -3 or x == 5

37.

4*x == 12-x^2

x^2+4*x-12 == 0

(x-2)*(x+6) == 0

x == 2 or x == -6

38.

x^2 == 15-8*x

x^2+8*x-15 == 0

x^2+8*x+16 == 31

(x+4)^2 == 31

x+4 == sqrt 31 or x+4 == -sqrt 31

x == -4+sqrt 31 or x+4 == -sqrt 31

x == -4+sqrt 31 or x == -4-sqrt 31

39.

x^2-9*x+18 == 0

(x-3)*(x-6) == 0

x == 3 or x == 6

40.

x^2+14*x+24 == 0

(x+2)*(x+12) == 0

x == -2 or x == -12

41.

(3*x+5)^2+(x-5)^2 == 40

9*x^2+30*x+25+(x-5)^2 == 40

9*x^2+30*x+25+x^2-10*x+25 == 40

⇒ algebra.equations.linear.merge

10*x^2+20*x+50 == 40

x^2+2*x+5 == 4

x^2+2*x+1 == 0

(x+1)^2 == 0

x+1 == 0

x == -1

42.

4*(10-x^2) == -2*x*(2*x+10)

2*(10-x^2) == -x*(2*x+10)

20-2*x^2 == -2*x^2-10*x

20 == -10*x

-2 == x

⇒ algebra.equations.linear.flip

x == -2

43.

x*(x+12) == 2*x^2

x^2+12*x == 2*x^2

-x^2+12*x == 0

-x*(x-12) == 0

x == 0 or x == 12

44.

3*(x-2)*(x+6) == 12*x

(x-2)*(x+6) == 4*x

x^2+4*x-12 == 4*x

x^2-12 == 0

⇒ algebra.equations.quadratic.no-lin

x^2 == 12

x == sqrt 12 or x == -sqrt 12

x == 2*sqrt 3 or x == -2*sqrt 3

45.

8*x^2+4*x-24 == (x+3)*(x-8)

8*x^2+4*x-24 == x^2-5*x-24

8*x^2+4*x == x^2-5*x

7*x^2+9*x == 0

x*(7*x+9) == 0

x == 0 or 7*x+9 == 0

x == 0 or 7*x == -9

x == 0 or x == -9/7

46.

3*x^2-11 == (3+2*x)^2

3*x^2-11 == 9+12*x+4*x^2

-x^2-12*x-20 == 0

(x+2)*(x+10) == 0

x == -2 or x == -10

47.

2*x*(x-3)-3 == (x+2)*(x+6)

2*x^2-6*x-3 == x^2+8*x+12

x^2-14*x-15 == 0

(x+1)*(x-15) == 0

x == -1 or x == 15

48.

12*(x^2-3*x)+8 == 56

12*(x^2-3*x) == 48

x^2-3*x == 4

x^2-3*x-4 == 0

(x+1)*(x-4) == 0

x == -1 or x == 4

49.

4*x^2-6*x == x^2+9

3*x^2-6*x-9 == 0

⇒ algebra.equations.quadratic.scale

x^2-2*x-3 == 0

(x+1)*(x-3) == 0

x == -1 or x == 3

50.

(x+1)*(x-5) == (x-2)*(x-3)

x^2-4*x-5 == x^2-5*x+6

-4*x-5 == -5*x+6

x-11 == 0

x == 11

51.

x^2+4*x-4 == 0

x^2+4*x+4 == 8

(x+2)^2 == 8

x+2 == sqrt 8 or x+2 == -sqrt 8

x == -2+sqrt 8 or x+2 == -sqrt 8

x == -2+sqrt 8 or x == -2-sqrt 8

x == -2+2*sqrt 2 or x == -2-2*sqrt 2

52.

x^2-6*x == 4

x^2-6*x-4 == 0

x^2-6*x+9 == 13

(x-3)^2 == 13

x-3 == sqrt 13 or x-3 == -sqrt 13

x == 3+sqrt 13 or x-3 == -sqrt 13

x == 3+sqrt 13 or x == 3-sqrt 13

53.

x^2-12*x+34 == 0

x^2-12*x+36 == 2

(x-6)^2 == 2

x-6 == sqrt 2 or x-6 == -sqrt 2

x == 6+sqrt 2 or x-6 == -sqrt 2

x == 6+sqrt 2 or x == 6-sqrt 2

54.

2*x^2+4*x-8 == 0

⇒ algebra.equations.quadratic.scale

x^2+2*x-4 == 0

x^2+2*x+1 == 5

(x+1)^2 == 5

x+1 == sqrt 5 or x+1 == -sqrt 5

x == -1+sqrt 5 or x+1 == -sqrt 5

x == -1+sqrt 5 or x == -1-sqrt 5

55.

(x-4)*(x-1) == 11

x^2-5*x+4 == 11

x^2-5*x-7 == 0

x^2-5*x+25/4 == 53/4

(x-5/2)^2 == 53/4

x-5/2 == sqrt (53/4) or x-5/2 == -sqrt (53/4)

x == 5/2+sqrt (53/4) or x-5/2 == -sqrt (53/4)

x == 5/2+sqrt (53/4) or x == 5/2-sqrt (53/4)

x == 5/2+1/2*sqrt 53 or x == 5/2-1/2*sqrt 53

56.

(x-7/2)^2 == 2*(x+4)

x^2-7*x+49/4 == 2*(x+4)

x^2-7*x+49/4 == 2*x+8

x^2-9*x+17/4 == 0

x^2-9*x+81/4 == 16

(x-9/2)^2 == 16

x-9/2 == 4 or x-9/2 == -4

x == 17/2 or x-9/2 == -4

x == 17/2 or x == 1/2

57.

x^2-3*x == 3*(x-2)

x^2-3*x == 3*x-6

x^2-6*x+6 == 0

x^2-6*x+9 == 3

(x-3)^2 == 3

x-3 == sqrt 3 or x-3 == -sqrt 3

x == 3+sqrt 3 or x-3 == -sqrt 3

x == 3+sqrt 3 or x == 3-sqrt 3

58.

(4-x)*(1-x) == 3*x

4-5*x+x^2 == 3*x

x^2-8*x+4 == 0

x^2-8*x+16 == 12

(x-4)^2 == 12

x-4 == sqrt 12 or x-4 == -sqrt 12

x == 4+sqrt 12 or x-4 == -sqrt 12

x == 4+sqrt 12 or x == 4-sqrt 12

x == 4+2*sqrt 3 or x == 4-2*sqrt 3

59.

2*x^2 == x*(x+2)+7

2*x^2 == x^2+2*x+7

x^2-2*x-7 == 0

x^2-2*x+1 == 8

(x-1)^2 == 8

x-1 == sqrt 8 or x-1 == -sqrt 8

x == 1+sqrt 8 or x-1 == -sqrt 8

x == 1+sqrt 8 or x == 1-sqrt 8

x == 1+2*sqrt 2 or x == 1-2*sqrt 2

60.

(1-x)^2 == x+2

1-2*x+x^2 == x+2

x^2-3*x-1 == 0

x^2-3*x+9/4 == 13/4

(x-3/2)^2 == 13/4

x-3/2 == sqrt (13/4) or x-3/2 == -sqrt (13/4)

x == 3/2+sqrt (13/4) or x-3/2 == -sqrt (13/4)

x == 3/2+sqrt (13/4) or x == 3/2-sqrt (13/4)