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And extract. the root, gives x2-a=±√a2+b;
Then transpos. a, gives xa2+b+a;

And extract. the root, gives x= ±√a±√a2+b. And thus, by always using similar words at each line, the pupil will resolve the following examples.

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1. To find two numbers whose difference is 2, and product 30.

Let x and y denote the two required numbers*.

Then the first condition gives x-y=2.

And the second gives xy=80.

Then transp. y in the 1st gives x=y+2;

This value of a substitut. in the 2d, is y2+2y=80

Then comp. the square gives y2+2y+1=81;

And extrac. the root gives y+1=9;

And transpos. 1 gives y=8;

And therefore x=y+2=10.

*These questions, like those in simple equations, are also solved by using as many unknown letters, as are the numbers required, for the better exercise in reducing equations; not aiming at the shortest modes of solution, which would not afford so much useful practice.

2. To divide the number 14 into two such parts, that their product may be 48.

Let x and y denote the two numbers.
Then the 1st condition gives x+y=14,
And the 2d gives xy=48.

Then transp. y in the 1st gives x=14-y;

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This value subst. for in the 2d, is 14y-y=48;
Changing all the signs, to make the square positive,
gives y3 14y=— 48 ;

Then compl. the square gives y2-14y+49=1;
And extrac. the root gives y−7=±1;

Then transpos. 7, gives y=8 or 6, the two parts.

3. Given the sum of two numbers 9, and the sum of their squares = 45; to find those numbers.

Let x and y denote the two numbers
Then by the 1st condition x+y=9.

And by the 2d x2+g2=45.

Then transpos. y in the 1st gives x=9-y;

This value subst. in the 2d gives 81 - 18y+2y2=45:

Then transpos. 81, gives 2y-18y=-36;

And dividing by 2 gives y29y=— 18;

Then compl. the sq. gives y9y+3=2;

And extrac. the root gives y― J = ± };

Then transpos. 2 gives y=6 or 3, the two numbers.

4. What two numbers are those, whose sum, product, and difference of their squares, are al! equal to each other?

Let x and y denote the two numbers.

Then the 1st and 2d expression give x+y=xy.
And the 1st and 3d give x+y=x2 — y2.

Then the last equa. div. by x+y, gives 1=x-y;

And transpos. y, gives y+1=x;

This val. substit. in the 1st gives 2y+1=y2+y;

And transpos. 2y, gives 1-y3 —y;

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Then complet the sq. gives y3 −y+} ;

And extracting the root gives 5y

And transposing gives 5+1=y;

And therefore x=y+1=1/5+1.

And if these expressions be turned into numbers, by extracting the root of 5, &c. they give x=2-6180+, and y=1.6180+.

5. There are four numbers in arithmetical progression, of

which the product of the two extremes is 22, and that of the means 40; what are the numbers?

Let x= the less extreme,

and y the common difference;

Then x, x+y, x+2y, x+3y, will be the four numbers.
Hence by the 1st condition x2+3xy=22,

And by the 2d x2+3xy+2y2

40.

Then subtracting the first from the 2d gives 2y= 18;
And dividing by 2 gives y2=9;

And extracting the root gives y=3.

Then substit. 3 for y in the 1st, gives x2+9x=22;
And completing the square gives x2 + 9x+® ] = 1 ‡ •;
Then extracting the root gives x+=1;

And transposing gives x=2 the least number.
Hence the four numbers are 2, 5, 8, 11,

6. To find 3 numbers in geometrical progression, whose sum shall be 7, and the sum of their squares 21.

Let x, y, and z, denote the three numbers sought.
Then by the 1st condition xz=y2,

And by the 2d x+y+z=7,

And by the 3d x2+y2+z2=21.

Transposing y in the 2d gives x+2=7-y;

Sq. this equa. gives x2 + 2xz +z2+=49—14y+y3 ;
Substi. 242 for 2xz, gives x2 + 2y2+22=49-14y+y2;
Subtr. ya from each side, leaves 18+y+2249-14y;
Putting the two values of x2+ y2+22
21-49-14y;

equal to each other, gives (

Then transposing 21 and 14y, gives 14y=28;
And dividing by 14, gives y=2.

Then substit. 2 for y in the 1st equa. gives xz=4,
And in the 4th, it gives x+z5;

Transposing z in the last, gives x-5-2;

This substit. in the next above, gives 5z-z2=4;

Changing all the signs, gives 22-52-4;

Then completing the square, gives 22-5z+25=2;

And extracting the root gives z={=±};

Then transposing gives z and x=4 and 1, the two other numbers;

So that the three numbers are 1, 2, 4.

QUESTIONS FOR PRACTICE.

1. WHAT number is that which added to its square makes

427

Ans, 6.

2. To

2. To find two numbers such, that the less may be to the greater as the greater is to 12, and that the sum of their squares may be 45.

3. What two numbers are those, whose the difference of their cubes 98 ?

Ans. 3 and 6. difference is 2, and

Ans. 3 and 5.

4. What two numbers are those whose sum is 6, and the sum of their cubes 72 ? Ans. 2 and 4. 5. What two numbers are those, whose product is 20, and the difference of their cubes 61; Ans. 4 and 5.

6. To divide the number 11 into two such parts, that the product of their squares may be 784. Ans. 4 and 7.

7. To divide the number 5 into two such parts, that the sum of their alternate quotients may be 41, that is of the two quotients of each part divided by the other.

Ans. 1 and 4.

8. To divide 12 into two such parts, that their product may

be equal to 8 times their difference.

9. To divide the number 10 into two such square of 4 times the less part, may be 112 square of 2 times the greater.

Ans. 4 and 8. parts, that the more than the

Ans. 4 and 6.

10. To find two numbers such, that the sum of their squares may be 89, and their sum multiplied by the greater may pro

duce 104.

Ans. 5 and 8.

11. What number is that, which being divided by the product of its two digits, the quotient is 54; but when 9 is subtracted from it, there remains a number having the same digits inverted? Ans. 32.

12. To divide 20 into three parts, such that the continual product of all three may be 270 and that the difference of first and second may be 2 less than the difference of the second and third. Ans. 5, 6, 9.

13. To find three numbers in arithmetical progression, such that the sum of their squares may be 56, and the sum arising by adding together once the first and 2 times the second and 3 times the third, may amount to 28. Ans. 2, 4, 6.

14. To divide the number 13 into three such parts, that their squares may have equal differences, and that the sum of those squares may be 75. Ans. 1, 5, 7.

15. To find three numbers having equal differences, so that their sum may be 12, and the sum of their fourth powers 962. Ans. 3, 4, 5.

16. To find three numbers having equal differences, and such that the square of the least added to the product of the two greater may make 28, but the square of the greatest added to the product of the two less may make 44.

Ans. 2, 4, 6.

17. Three

17. Three merchants, A, B, C, on comparing their gains find, that among them all they have gained 14441.; and that B's gain added to the square root of A's made 9201.; but if added to the square root of c's it made 912. What were

their several gains?

Ans. A 400, в 900, c 144.

18. To find three numbers in arithmetical progression, so that the sum of their squares shall be 93; also if the first be multiplied by 3, the second by 4, and the third by 5, the sum of the products may be 66. Ans. 2, 5, 8.

19. To find four numbers such, that the first may be to the second as the third to the fourth; and that the first may be to the fourth as 1 to 5; also the second to the third as 5 to 9; and the sum of the second and fourth may be 20.

Ans. 3, 5, 9, 15.

20. To find two numbers such that their product added to their sum may make 47, and their sum taken from the sum of their squares may leave 62. Ans. 5 and 7.

RESOLUTION OF CUBIC AND HIGHER
EQUATIONS.

A CUBIC Equation, or Equation of the 3d degree or power, is one that contains the third power of the unknown quantity. As x3-ax2+bx=c.

A Biquadratic, or Double Quadratic, is an equation that contains the 4th Power of the unknown quantity :

As x-ax3+bxa — cx=d.

An Equation of the 5th Power or Degree, is one that contains the 5th power of the unknown quantity:

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