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17

15 tracting the root, it.

:t and by transposition, x=-8,

4 or -

Ex. 5. Given 4x _3x85, to find the values of x.

(Art. 417). Multiplying by 16, 6402—48x=1360, and, adding the square of 3, 64x2448x+9=1369; ... extracting the square root, 8%=3=+37; by transposition, 8x=40, or -34, ...=5, or-41.

35-3x Ex. 6. Given 6x+ -=44, to find the values of x. Multiplying by 1, 6x +35-3x = 442 ; by transpositioa, 6x4_47x=-36 ;

47

35 and by division, za

; therefore completing the 6

6
47 47 2209 30 1369

i.. extracting 12 144 6

144
47 37
the root, 2 -

and x=7, or á.
12 12'
3.3

3x -6 Ex. 7. Given 5x - =2x+ to find the values

square, ca

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3

of 2.

Multiplying by 2x-6, we have 10.x2-36x+6=4x2 - 12x +33 -15%+18; .. by transposition, 3x2_9x=12 ;

and by division, 22_3r=4 ::
... completing the square, x2-3x+=4+1=,
and extracting the root, x-r+;
.. x=4, or-.

3x - 10 6x240
Ex. 8. Given 3.0

=2+ -, to find the va9- 2:30

2:11 lues of x. Multiplying by 2x-1,

6x4-23x+10 6x3x

=4x-2 +6x-40,
92x
6.22–23x + 10

9- 2x
63.x— 14x2 +6x2_23x+10=378-84%;
by transposition, 124x8x2=368

and 2~34x=446 ;.. by completing the square, 31 961 961

225 ct

-46= 2

i 16

16

31 15 ... extracting the root, 2am.

i

or 7x-+

=42 ;

xa

16

n

n

23 and therefore x=

or 4.

2 Ex. 9. Given +x=6_x, to find the values of x.

Dividing by Vx, 2+x=6: .. completing the square, x+x+1=6+1=; and extracting the root, x+1=+;

:'. x=2, or-3. 'Ex. 10. Given xn_2ax?=b, to find the values of x.

=. Completing the square, x*—2ax? +a?=a?+b; .. extracting the root, æž -a=+v(a+b), andzi=av(ao+b); :: x=(«+(a+6)).

Ex. 11. Given 22–2x+6V (72_2x+5)=il, to find the values of x. Adding 5 to each side of the equation,

(22-22-+5)+6_(QP-2x+5)=16; .. by completing the square,

(2_2x+5)+6/(2* -- 2x+5)+9=25 ; and extracting the root, (x2_2x+5)+3=5; .: (x2–2x+5)=2, or -8; ..squaring both sides, x2–2x+5=4, or 64 ; whence za 2x+1=0, or 60 ; and extracting the root, 1-1=0, or #60;

.. r=1, or 1+ / 60. Schol. It is proper to observe, that the equation, 29- 2x +1, has two equal roots, although x appears to have only one value ; but it is because x is twice found =1, as the common method of resolution shows ; for we have x=1+70, that is to say, & is in two ways=1.

Ex. 12. Given x*+ 4x +12° +16x=a, to find the values

of x.

Here the two first terms of the square root of the left-hand member (Art. 299), is found to be 32+2x, and the remainder is 8x2 +-16x which can be readily resolved into the factors 8 and x2+2x, since (8x2 +16x) =(22+2x) gives 8 for the quotient. Consequently the proposed equation may be exhibited under the quadratic form (x2 +22)2 +8(x2+2x)=a ; ... by completing the square, (r2 +2x) +8(:r2+2x)+16=a+ 16 ; and extracting the root, 22 +23+4=;/(a +16) Now by taking the positive sign,

2* + 2x+4t(a+16) ; by transposition, 22+2x=-4+v(a+16); is completiog the square, x+2:ti=-3+v(a+16);

and extracting the root, x+1=+v(3+(a+16));

.: x=-1/3+(a+16). Again, by taking the negative siga.

x +2+4=-Va+16); ... 22+2x=-4-(a +16); and completing the square, +20+1=-3-v(a+16); ... extracting the root, x+1=+v(-3-Va+16));

and x=-1+7(-3-va+16)). Ex. 13. Given 3x2–2x+12=16—4, to find the values

of x.

By transposition, 3x2—12x=16—4—12=0;

and by division, 24--4x=0; ... by completing the square, x*— 4x+4=4, and extracting the root, x-2=+2 ; ..x=4, or 0.

See (Art. 405). Ex. 14. Given xi-40° +60=4, to find the values of x.

(Art. 423), multiplying both sides by x, 24–4x3+6x24x 0,

(Art. 422) .:(22_2x)2 + 2(22-2x)=0.

..m-2.0+1=+1, and x=1+1+1; :. the three roots of the proposed equation, are 1, 1+-1, and 1-V-1. The other value of x, which is equal to 1-1, or 0, belongs to the equation (-2x)+2(x-2x)=0; hence there are four roots, or four values of x, which will satisfy this last equation.

841 17 232 1 Ex. 15. Given 27xo +

-+5, to find the

322 3 32 3x2 values of x. Multiplying every term by 3,

841

232 1 812 -+175

xc3

1 841 232 .. by transposition, 81x2 +17+.

+ +15.

2012 22 Adding unity to each side, in order to complete the square ;

1 841 232 312° +18+ -t

+15;

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-+16;

22

1

29

and extracting the root, 9x+=+6+4). Let the positive value be taken ; then by transposition, 9x

, and ... 9x - 4x=28; by completing the square, &c.,

28

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14

we shall have x=2, or

But if the negative value be

.

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taken, 9x2 +40=-30 and completing the square, &c., = -2+v1-266)

9 Ex. 16. Given 3x2 + 2x-9=276, to find the values of x.

17 Aos. x=5, or

3 8

2x-11 -2 Ex. 17. Given

to find the values 2

-3 6 of x.

Ans. x=6, or 3.0 +04 30-2, 73-14 Ex. 18. Given

to find the values 5 X-6 10 of x.

Ans. x=36, or 12. 23-10x + 1 Ex. 19. Given

= -3, to find the values of x. -6x+9

Ans. x=-1, or-28. Ex. 20. Given (x+5) XV (x+12)=12, to fiud the values of x.

Ans. x=4, or -21. Ex. 21. Given 2x+3x-57(262+3x+9+3=0, to find

9 the values of r. Ans. a=3, or

- 3+/-55 or 2

4 Ex. 22. Given 9x +(16x9 +362") = 152 – 4, to find the

4 1 9+481 values of x.

Ans. x=

3

50 49.x2 48

6 Ex. 23. Given

+

· 49=9+-, to find the values 4

8 -3+93 · Ans. x=2, or

7

7 Ex. 24. Given x4—2x +-x=132, to find the values of x.

1+1(-43) Ans. x=4, or --3, 0.

2 Ex. 25. Given aš taš=756, to find the values of x.

Ans: x=243, or (-28) Ex. 26. Given x2-x2=56, to find the values of x.

Ans. r=4, or (-7). Ex. 27. Given x+5=V (x+5)+6, to find the values of x.

Ans. x=4, or -1. Ex. 28. Given <+16-77 (0+16)=10—4(:+16), to find the values of x.

Ans. x=9, or —

-12. 72-8 Ex. 29. Given x +4+. =13, to find the values of x.

13

or

i or

22

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of x.

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or

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Aos. =4, or

+42_2+7

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of 2.

9+42 Ex. 30. Given 144-4x. = 3:1 + to fiod the va

3 lues of x.

Ans. x=28, or 9. ..+4 1- 4x +7 Ex. 31. Given

-1, to find the values 3

3 9 of x.

Ans. x=21, or 5. 8x +16

123 - 11 Ex. 32. Given 2x +18

=27

to find 4x+7

21-3 the values of x.

Ans. x38, or 5. **+2x+8 Ex. 33. Given

=+2+8, to find the values +x-6

Ans. x=4, or -4. Ex. 34. Given ✓(4x+5) XV(70+1)=30, to find the values of x.

Ans. d=5, or -631. x +12

78 Ex. 35. Given

to find the values of x. a +1215

Ans. x=3, or -15. Ex. 36. Given tš +7.7% =44, to find the values of x.

Ang. x=+8, or +(Ex. 37. Given 421+x==39, to find the values of x.

13

Ans. x=729, or Ex. 38. Given 3x® +42x3=3321, to find the values of x.

Ans. x=3, or —41. 8 17 Ex. 39. Given +2= to find the values of x.

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+(-11)

()

of x.

Ans. x=4, or iv2. Ex. 40. Given x +11+(x3 +11)=42, to find the values

Ans. x=+5, or 38. Ex. 41. Giveo x*— 12x +-50=0, to find the values of x.

Ads. r=6+v(-14). Ex. 42. Giveo 3x~7*2=10, to find the values of x.

Ans. I=6+/-4. Ex. 43. Given æ~2.03=48, to find the values of x.

Ans. x=2, or V-6. Ex. 44. Given x4 +-222-722 - 8x3-12, to find the values of x.

Ans. 2, or -3, or 1, or —2. Ex. 45. Given 29-10x?+3522-50.3+24=0, to find the values of x.

Aos x=

- 1, 2, 3, or 4. Ex. 46, Given 23 - 8x8 + 19.2-12=0, to find the values

Ans. x=1, 3, or 4.

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