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22. 6x2 - 8 /320* + 5 x 4 12. 23. 2 + + 5x – 2 x 22 - 7x + 1 - 35. 24. 2022

8x

N/5 20 3 ax + 2 ao = = 7 a?. 25. a*(x - 2) + 6 x*(x - 2) = 24 00 + 36 - 5x?. 26. 2 x + 20 x 13

22 2C + 5 = 105. 27. 4x + 12 x – 2 x 14 x* – 2 x + 19 = 30, 28. a* + ac* + 1 = a(a + x + 1), 29. 2 1 0.

3 (362 31. (3C

30. (20 – a)3 + (2x2 – 6)3 + (2x2 – c):

a) (x2 - b) (22 – c) + 9 axé a(a + b + c)?.

b) (x2 – c) = (a - b) (ao – co). 32. (x – a) (x – 5) (x – c) = (m – a) (m b) (m - c). 33. (a a) (2 b) (x – c) (a + b)(a + c)(b + c). 34. (12

1)ą (ac? + ax + a) = a* - 20 35. (ax 3)+ (cx d)* = (a + c) acs – (6 + d)?. 36. 22" +22"="+ 1 = a (20* + x + 1).

Simultaneous Quadratic Equations.

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6. The following worked examples are given as specimens of the methods to be employed, but it must be understood that practice alone will give the student complete mastery over equations of this class. Ex. 1. Solve xa + yo 20..... 2 + y 6........

(2.) As we have given the sum of the unknown quantities, we shall work for the difference. From (2), multiplying each side by 2, we have

2 22 + 2y2

40 and from (1), squaring,

ana + 2 xy +

= 36 Then, subtracting,

aca
2 xy +

ya 4; and, taking the square root, we have x - y = + 2........(3).

(2) + (3), then 2 x 8 or 4, and : 4 or 2. (2) (3), then 2 y 4 or 8, and .: Y = 2 or 4.

NOTE.—Having found that x = 4, y 2, we might have told by inspection that the values x = 4, would also satisfy the given equations, for x and y are similarly involved in both equations. Ex. 2. Solve ac? ya 5........

(1.)
ху
= 6.

(2.) As we have given here the difference of the squares of the unknown quantities, it will be convenient to work for the sum of the squares,

2, y

9 or

=

8 or and :: Y

11 3....

11 y.

From (1.), squaring, - *y

2 xʻyo + y4 25, and from (2.), squaring, &c., 4

144 Then, adding,

XA

+ 2 x*yo + y = 169 and taking the square root, x2 + y2 = + = 13.......(3). (3.) + (1.), then, 2 xc = 18 or 8, or x

4, and ... X =

+ 3 or + 2 - 1. (3.) - (1.), then, 2 ya

18, or y = 4 or 9,

† 2 or † 3 - 1. NOTE.—The student will see that the pairs of values which satisfy the given equations are, x = 3, y = 2; x = - 3, y = -2; x = 2N - 1, y = 3V-1; x = 2V - 1, y =

- 3V - 1. Ex. 3. Solve ac + y

(1.) ? ya + x =

(2.) S. Subtracting, then, a* - y - x + y = 11 x - 11 y;

or,

22 ya 12 ( - y). Now (wc y) is a factor of each side, and hence, striking it out, we have x + y = 12 ......

.(3.), and also a

(4.). Equations (3.) and (4.) may not be used as simultaneous equations, but each of them may be used in turn with either of the given equations. Thus, taking equations (3.) and (1.), we have

(1.) – (3.), ac

from which x = 6 + 2 V6; and hence from (3.), by substitution, we easily get

y = I
Again, taking equations (4.) and (1.)—

we have, from (4.) ® Y,
and :: from (1.), 3c2 + x 11 x or coa

y = 0........

X =

11 ac

12,

from which x = 10 or 0; and so, from (4.), y = 10 or 0.

10 x,

Hence, the pairs of values satisfying the given equations

are

X =

0, y

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14 xy ;

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88 y.

10, y = 10; x = 0, y = 0; 2 = 6 + 2 6, 9 = 6 – 9 6 ;

c = 6 - 2 N6, y = 6 + 9/6. NOTE.—It is worth while remarking that when each of the terms of the given equations contain at least one of the unknown quantities, the values x = O will always satisfy. Ex. 4. Solve 3 *

55.

(1.) / 2 - 5 xy + 8 y = 7

(2.) | Multiplying the equations together crosswise, we get

55 22 275 xy + 440 ya = 21 cm or, transposing, 34 ** - 261 xy + 440 y* = 0; or, (2 x

5 y) (17 a 88 y) = 0, from which 2x 5

Y,

and 17 x = Each of these equations taken in turn with either (1.) or (2) will easily give the required values of x and y. Ex. 5. 204 + y4

(1.), + y 7

(2.) From (2), raising each side to the fourth power, we have

Oct + 4x*y + 6 x*y* + 4 xy + y = 2401 ;.. .(3.) (3) - (1), then 4 xy + 6 x*ya + 4 xy = 2064; or, 2xy + 3x*y* + 2xy

1032; or, arranging, 2 xy(x + y) - x*yo = 1032; but from (2), (x + y)?

49, and hence, 2 wy(49) - a*y*

1032; 98:

0, from which xy = 12 or 86,...... (4.) From (2) and (4), x - y may now be easily obtained, and hence also the required values of x and y.

337......

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27, wy

= 30.

+

+

ya = 6.

5. ** + y2 = 29, x y 6. Bc2 ya = 13, (OC

13, (x - y) = 1. 7. ac – ya

18. 8. x2 ya = 12, x + y = 6. 9. ac + y a

53, * - y = - 45. 10. 2a + xy 28, y + xy

21. 11. zca + xy + ya

19, xy

ocia 3. 12. x + y

13, Vac + Vy = 5. 13. x2 + xy + ya 84, c + 2y + y = 14. 14. 203 + y2 = 35, wʻy + xya

1 1 1 1 15. a,

6. Y

ya 16. 3 + Y

a( y), ac+ y = 62. 17. och y = a, 22 18. 2 + y

5,203 + 18 35. 19. x + y 5, 205 + ys

275. 20. ** + y'=Y (2 + y), xy = 6.

Y

2, as - 1 22. xy (20 + y) = a, ** y* (2 + y) = b. 23. xy(x + y) = 30, a?y*(c + y) 9900. 24. 4 cm

18, 5 ya

30 25. a* + y} = 35, aż + y

xły
26. a1 + y$ = 3 x, x1 + y}
216

77
3C + y + 4
xy

x + y 28. (c + y)2 + 2(x – y)2 = 3 (2c + y) (x

3 (x + y) (x y), sc2 + y = 10. 29. ** + 10 xy + y = ? (ac – y), a + 5y = a + 13 y. 30. ** – 2 xya + y = 1 + 4 xy, a* (x + 1) + y(y + 1) = sy.

9 x2 ya 117 8.2 - y2 + 1 31. 3.c + y - 9 =

6 4x + y + 1

21. X

98.

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