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EXAMPLES.

1

55 ;

1. Given xo + 43 = 60; to find x.

First, by completing the square, xo + 4x + 4 = 64 ;
Then, by extracting the rooi, t + 2 = = I 8;

Then, transpos, 2, gives I = 6 or 10, the two roots., 2. Given a 6x + 10 = 65 ; to find x.

First trans. 10 gives a ? 62
Then by complei. the sq. it is .r? 60 + 9 = 64;
And by extr. the rooi, gives r 3 = 8;

Then trans. 3, gives x = 11 or 5.
3. Given 2x2 + 8x 30 = 60; to find x.

First by transpos. 20, it is 2x2 + 8x = 90;
Then diy, by 2, gives x2 + 4x = 45 ;
And by compl. the sq. it is x2 + 4.3 + 4 = 49;
Then extr. the root, it is r + 2 = +7;

And transp 2, gives x = 5 or 9.
4. Given 3x? 3x + 9 = 8}; to find s.
First div by 3, gives x2

x + 3

27;
Then transpos. 3, gives x?
And compl. the sq. gives x? . x + = t;
Then extr. the root gives x

And transp. 1, gives x = - or } 3. Given tr3

- 4x + 301 52 ; to find x,
First by transpos, 305, it is sx? - for = 22;
Then mult by 2 gives 2: = 441;
And by compl. the sq. it is x? fix +
Then extr. the root, gives x - } + bý;

And transp. , gives x = = 7 or 61 $. Given ar? bx sc; to find x.

6 First by diy. by a, it is x?

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a

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6 6 69 Then compl. the sq. gives - 2+

+-; 422

42 6

4ac + 6? And extrac. the root, gives x

3
20
6

4ac + 6? 6
Then transp.-
gives = tr

+ 2a

42%

2a 4, Given ** 2ax2 = 0; to find x. First by compl. the sq. gives **- 20x2 + 2 = ' +6;

And

And extract. the root, gives x2 -a= I Va? + b;
Then transpos. a, gires x2 = + val + 6+a;

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

EXAMPLES FOR PRACTICE.

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l. Given x2

60

7 = 33; to find x, Ans. ¢ = 10 2. Given ?

10 = 14 : to find x. Ans, = 8. .3. Given 5.x2 + 4x - 90 = 114 ; to find x. Ans. x = 6. 4. Given jx2 - + 2 : 9; to find x. Ans. X = 4. 5. Given 3. * - 2x2 = 40; to find x.

Ans x=2. 6. Given fot -- IV x = l; to find .

Ans. x = 9. 7. Given jx2 +, for ; to find x. Ans. x = 727766. 8. Given x6 + 4x3 = 12; to find x.

Ans. Is

2=1.259921. 9. Given x2 + 40 = a + 2; to find x.

Ans.x = V2 + 6-2.

QUESTIONS PRODUCING QUADRATIC EQUATIONS.

Y = 2,

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

duct 80.
Let x and y denote the two required numbers*.

Then the first condition gives x -
And the second gives xy = 80.
Then transp. y in the ist gives < = y + 2;
This value of x substitut in the 2d, is yế + 2y = 80
Then comp. the square gives ya + 2y + 1 = 81,
And extrac. the root gives y +1=9;
And transpos. I 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 Ist condition gives x + y = 14,
And the 2d gives xy = 48.
Then transp. y in the 1st gives x = 14
This value subst. for r in the 2d is 147 - y? 48;
Changing all the signs, to make the square pusi uve,

gives ya 14y = -
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.

Y;

48 ;

their squares

3. Given the sum of two numbers = 9, and the sum of

= 45; to find those numbers.
Let x and y denote the two numbers.
Then by the 1st condition x + y
And by ihe 2d x2 + y2 = 45
Then transpos y in the 1st gives x = 9-y;
This value sulis in the 2d gives 81 18y + 2y = 45;
Then transpos. 81, gives 242 8y
And dividing by 2 gives ya 9y
Then compl. the sq. gives y

92 +
And extrac. the root gives y
Then transpos i gives y = 6 ur 3, the two numbers.

36;

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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 it + y = xy,
And the 1st and 3d give x +y

y%.
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 = y + y;
And transpos. 2y, gives I = y2 - Y;
Then complet. the sq. gives - y - yti
And extracting the root gives V 5=y
And transposing I gives V 5+5=Y;

And therefore x = y +1=5+
And if these expressions be turned into numbers, by ex.

tracting the root of 5, &c. they give x = 2:6180 +, and

y = 16180 +. -5. There are four numbers in arithmetical progression, of

which

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

Let I = the less extreme,

and y = the common difference ;
Then x, s+ y, x + 2y, x + 3y, will be the four numbers.
Hence by the 1st condition x? * 3 ry

- 22,
And by the 2d r+ 3xy + 2y2 = 40,
Then subtracting the first from the 2d gives 2ye = 18;
And dividing by 2 gives y? = 9;
And extracting the root gives y s 3.
Then substit. 3 for y in the 1st, gives xl + 9.3 = 22 ;
And completing the square gives r? + 9x + 6°;

Then extracing the root gives r+=;
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 = y?,
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 x? + 2.cz + 2 + 49 14y + y;
Substi. 2y2 for 2x2, gives x2 + 2y2 + z2 = 49 - 14y + y? ;
Subtr. ya from each side, leaves x2 + y2 + z2 = 49–14v;
Putting the two values of x2 + y2 + z??

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 sz = 4,
And in the 4th, it gives x + z = 5;
Transposing z in the last, gives x = 5
This substit. in the next above, gives 52 224;
Changing all the signs, gives 22 - 5z = - 4;
Thin completing the square, gives z2 – 5z + 3 =
And extracting the root gives z otti
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

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

Ans. 6.

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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.

Ans. 3 and 6. 3. What two numbers are those, whose difference is 2, and the difference of their cubes 98 ?

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 45, that is of the two quotients of each part divided by the other.

Ans. I 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 iwo 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. il. 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 co the product of the two less may make 44.

Ans. 2, 4, 6.

17. Three

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