1. Given

EXAMPLES.

+ 4x = 60; to find x.
First, by completing the square, x2 +
Then, by extracting the root, ≈ + 2 =
Then, transpos. 2, gives x = 6 or
2. Given x2 6x + 10 = 65; to find x.

4x + 4 = 64; ± 8 ;

10, the two roots..

6x+9= 64;

First trans. 10 gives a2 6x = 55;
Then by complet. the sq. it is r2
And by extr. the root, gives x →→→
Then trans. 3, gives x = 11 or

3. Given 2x2 + 8x

-

3 = ± 8;

5.

3060; to find x.

First by transpos. 20, it is 2x2 + 8x = 90 ;
Then div. by 2, gives x2 + 4x = 45 ;

And by compl. the sq. it is x2 + 4x + 4 = 49;
Then extr. the root, it is +2 = ±7;

And transp 2, gives x = 5 or 9.

4. Given 3x2

3x+9= 8; to find x.

First div by 3, gives x2
Then transpos. 3, gives x2
And compl. the sq. gives x2
Then extr. the root gives x -
And transp., gives x = 3 or }.
5. Given 3 1x + 30}
First by transpos. 30), it is 2
Then mult by 2 gives x1

=

52; to find x.

- zx = 3x=44};

221;

And by compl. the sq. it is x2 - 3x + } = 44; ;

Then extr, the root, gives ≈

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

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

duct 80.

Let x and y denote the two required numbers*.

Then the first condition gives x

And the second gives xy = 80.

y = 2,

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

This value of ≈ 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;

8;

And transpos. 1 gives y =
And therefore x = y + 2 = 10.

pro

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 transpy in the 1st gives x = 14 -Y;
This value subst. for x in the 2d is 14y

y2 = 48;

Changing all the signs, to make the square positive,

gives y2

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.

4. What two numbers are those. whose sum, product, and difference of their squares, are all 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;

=

y2+y;

This val. substit. in the 1st gives 2y + 1
And transpos 2y, gives 1 = = y2 — y ;
Then complet. the sq. gives y2 −y + 1;
And extracting the root gives√5y;
And transposing gives√ 5+1 = y;
And therefore x = y + 1 = } √ 5 + 2/2.

2.6180+, and

And if these expressions be turned into numbers, by extracting the root of 5, &c. they give x y=16180 +.

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

and y

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 242 = 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 + 3
Then extracting the root gives x + 2 = 13:
And transposing gives x = 2 the least number.
Hence the tour numbers are 2, 5, 8, 11,

169 ;

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 + z = 7-y;

Sq. this equa. gives x2 + 2xz + z3 +

= 49 14y+ y2;

Substi. 2y2 for 2xz, gives x2 + 2y2 + z2 =49-14y+ y2;
Subtr. y2 from each side, leaves x2 + y2 + z2 — 49 — 14ʊ;
Putting the two values of x2 + y2 + z2

equal to each other, gives 21 = 49-14y;

S

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 + z =

5;

=

Transposing z in the last, gives x = 5 — z;
This substit. in the next above, gives 5z z2 = 4;
Changing all the signs, gives z2 — 5z — — 4;
Then completing the square, gives z25z+ 25 = 12;
And extracting the root gives z = ± };
Then transposing gives z and

other numbers;

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

3;

4 and 1, the two

QUESTIONS FOR PRACTICE

1. WHAT number is that which added to its

42?

square makes

Ans. 6.

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 and the difference of their cubes 98?

Ans. 3 and 6. difference is 2, 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 4, 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.

Ans. 4 and 8.

9. To divide the number 10 into two such parts, that the square of 4 times the less part, may be 112 more than the square of 2 times the greater. 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 produce 104. Ans. 5 and 8.

11. What number is that, which being divided by the product of its two digits, the quotient is 5; 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 the 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 3 times 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