32.

7x

√7x-5+√4x-1 = √7x − 4 + √4x − 2.

33. √a2-x+√b2 + x = a + b.

35. Ja-bx+√c-dx = √ a + c − (b + d) x.

39.

√(a + x) (x + b) + √(a − x) (x − b) = 2 √ax.

40. √a (a+b+x) − √a (a + b − x) = x.

41.

42.

/c

x2 + ax + a2 + √ x2 – ax + a2 = √2a2 – 2b2.

43. √ax-b+ √cx + b = √ ax + b + √

44.

45.

46.

cx - b.

x (a + b − x) + √ a (b + x − a) + √b (a + x − b) = 0.

x+a+ x+b+√x + c = 0.

√ab (a + b + x) = √ a (a + b) (b − x) + √b (a + b) (a − x).

47. √x2 - b2 - c2 + √ x2 - c2 − a2 + √√ x2 − a2 − b3 = x.

48.

√a3 − x3 + √b3 − x2 + √ c2 − x2 = √√ a2 + b2 + c2 − x3.

49. For what values of x is 14 – (3x − 2) (x − 1) real.

53. Find the greatest and least real values of x and y which satisfy the equation

x2 + y2 = 6x - 8y.

54. Find the greatest and least real values of x and y when

55. When x and y are taken so as to satisfy the equation (x2 + y2)2 = 2a3 (x2 — y3), find the greatest possible value of y.

56. Shew that if the roots of the equation

12

x2 (b2+b'3) + 2x (ab + a'b') + a2 + a'2 = 0

be real, they will be equal.

57. If the roots of the equation ax2 + bx + c = 0 be in the ratio mn, then will mnb2 = (m + n)2 ac.

58. If ax2+2bx + c = 0 and a'x2 + 2b'x + c = 0 have one and only one root in common, prove that b3 - ac and b’2 – a'c' must both be perfect squares.

59. If x, x be the roots of the equation ax2 + bx + c =

2

= 0,

2

find the equation whose roots are (i) x ̧3 and x ̧3,

3

3

хі

(ii)

and 20

(iii) b+ax, and b + a¤ ̧·

60. If x,, x, be the roots of ax2 + bx + c = = 0, find in terms of

a, b, c the values of :

61.

Shew that, if X1, X2 be the roots of x2 + mx + m2 + a = 0, then will x + x2x2 + x22 + α =

62.

2

0.

If x, x, be the roots of (x2 + 1) (a2 + 1) = max (ax − 1), then will (x2 + 1) (x2 + 1) = mxx ̧ (×× ̧ − 1).

Equations of higher degree than the second.

134. We now consider some special forms of equations of higher degree than the second, the solution of the most general forms of such equations being beyond our range.

135. Equations of the same form as quadratic equations.

can be solved in exactly the same way as the quadratic equation

we therefore have

ax2+ bx + c = 0;

Thus there are four real or imaginary roots.

Similarly, whenever an equation only contains the unknown quantity in two terms one of which is the square of the other, the equation can be reduced to two alternative equations: for, whatever P may be,

Thus there are four roots, namely + 1, − 1, +3, −3.

Ex. 2. To solve (x2+x)2+4 (x2+x) − 12=0.
The equation may be written

whence we obtain two values of y, a and ẞ suppose.

and the four roots of the last two quadratic equations are the roots required.

Ex. 6. To solve

1

3; -1; 1±√61.

9

(x+a) (x+2a) (x +3a) (x+4a)=16 a1.

Taking together the first and last of the factors on the left, and also the second and third, the equation becomes of the form we are now considering. We have