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determined, making the root of the known side either + or - , which will give two roots of the equation, or two values of the unknown quantity.

Note, 1. The root of the first side of the equation, is always equal to the root of the first term, with half the co-efficient of the second term joined to it, with its sign, whether'+ or -.

2. All equations, in which there are two terms including the unknown quantity, and which have the index of the one just double that of the other, are resolved like quadratics, by completing the square, as above. Thus, x4 + ax? =

ab,

or 22n + axh = b, or r + ax? 3b, are the same as quadratics, and the value of the unknown quantity may be determined accordingly.

Therefore, when x2 ax. =b, we shall have x = t.Vb+ 9 + ja for the affirmative value of x; and x = vbitu?+ a for the negative value of x; so that in both the first and second forms, the unknown quantity has always two values, one of which is positive, and the other negative.

But, in the third form, where x = # b. + a, both the values of x will be positive, when {a? is greater than b. For the first value, viz, =+vab + a will then be affirmative, being composed of two affirmative terms.

The second value, viz. x - VIQ? - b + ja is affirmae tive also; 'for since {ais greater than 12% b, therefore vaor ja is greater than via

- b; and consequently - vla - 6+ La will always be an affirmative quantity. So that, when x2 – ux =- b, we shall have r = + vad 6 + ja, and also x = Vlu -6+ Zu; for the values of r, both positive.

But in this third form, if b be greater than 1 a?, the solution of the proposed question will be impossible. For since the square of any quantity (whether that quantity be affirmative or negative) is always affirmative, the square root of a negative quantity is impossible, and cannot be assigned. But when b is greater than Lu”, then ja?'- b is a negative quantity; and therefore its root va -b is impossible, or imaginary; consequently, in that case, x = a=N— b, or the two roots values of t, are both impossible, or imaginary, quantities. VOL. I.

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EXAMPLES

EXAMPLES.

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1. Given era + 4x = 60; to find x.

First, by completing the square, .x? + 4x + 4 = 64; Then, by extracting the root, x + 2 = $ 8;

Then, transpos. 2, gives x = 6 or – 10, the two roots. 2. Given r? – 6x + 10 = 65; to find z.

First, trans. 10, gives z' - 6.r = 55;
Then by complet. the sq.it is x? – 6.x + 9 = 64;
And by extr. the root, gives x 3 = £8;

Then trans. 3, gives x = 11 or -5.
3. Given 2x2 + 8.0 - 30 = 60; to find r.

First by transpos. 20, it is 2.1? + 8.c = 90;
Then div. by 2, gives .r' + 4x = 45;

And by compl. the sq. it is 2.2 + 4.3 + 4 = 49;
-Then extr. the root, it is x +2=+7;

And transp. 2, gives x = 5 or — 9. 4. Given 3x – 3x + 9

8; to find .r.
First div. by 3, gives x?_X + 3 = 25;
Then transpos. 3, gives r--*=-
And compl. the sq. gives x-r+1 = ;
Then extr. the root gives x - Etti

And transp. *, gives x = ý or :
5. Given ir? - *x + 301 = 52}; to find x.

First by transpos. 30, it is r? - fr = 225;
Then mult. by 2 gives x? - žir = 444;
And by compl. the sq. it is r-— *x + y = 444;
Then extr. the root, gives'x — } = + 6 ;

And transp. , gives x = 7 or -63. 6. Given ar? - bx = c; to find .r.

b
First by div. by a, it is ti?

6
62

62 Then compl.the sq. gives x? X +

+

4a? 6

4ac + bi And extrac. the root, gives x =

2a

4a? b.

4ac + 62

b
2a
gives t = IN

+
4a?

2a 7. Given x4 2ax = b; to find r. First by compl. the sq. gives 24 - 2ax? + a2 = a +b;

And

a

a

с

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a

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Then transp

And extract. the root, gives x -a = Iva + b;
Then transpos. a, gives x2 = + a + b +a;

And extract. the root, gives x = + Vad + 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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1. Given r? – 6.3 - 7 = 33; to find x. Ans. = 10. 2. Given x2 5x - 10 = 14; to find x.

Ans. I = = 8. 3. Given 5.x2 + 4.2 - 90 = 114; to find r. Ans. X = 6. 4. Given {x?-12 + 2 = 9; to find r. Ans. x = 4. 5. Given 3.r4 _ 2x? = 40; to find x. Ans. .X = 2. 6. Given x - *V x = 1*; to find x. Ans. r = 9. 7. Given 1.x2 + çox = ; to find x. Ans. x = 727766. 8. Given 200 + 4x3 = 12; to find z.

Ans. t = 32 = 1*259921. 9. Given x2 + 4x = a + 2; to find x.

Ans. * = va+6- 2.

QUESTIONS PRODUCING QUADRATIC EQUATIONS.

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

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

Then the first condition gives x-y = 2,
And the second gives zy = 80.
Then transp. y in the 1st gives x=y + 2;
This value of r substitut. in the 2d, is y? + 2y = 80;
Then comp. the square gives y' + 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 aim, ing at the shortest modes of solution, which would not afford so much useful practice.

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2. To

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 * + y = 14,
And the 2d gives xy = 48.
Then transp. y in the 1st gives « = 14-Y;
This value subst. for x in the 2d, is 14-y = 48;
Changing all the signs, to make the square positive,

gives y: -- 14y = - 48;
Then compl. the square gives yo – 14y + 49 = 1;
And extrac. the root gives y7 +l;
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 r and y denote the two numbers.
Then by the 1st condition x + y = 9.
And by the 2d x2 + y2

= 45.
Then transpos. y in the 1 st gives * = 9-y;
This value subst. in the 2d gives 81 – 18y + 2y = 45,
Then transpos. 81, gives 2y – 18y = -36;
And dividing by 2 gives y-9y = -18;
Then comph the sq. gives y’ -- 9y + = 4
And extrac. the root gives y— = } };
Then transpos. gives y = 6 or 3, the two 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 l'st and 2d expression give x +y = xy,
And the 1st and 3d give x + y = *-*.
Then the last equa. div. by x + y, gives 1 = *-y;
And transpos. y, gives y + 1 = x;
This val. substit. in the 1st gives 2y + 1 = gi + y;
And transpos. 2y, gives 1 = y-y;
Then complet. the sq. gives ge--y-*;
And extracting the root gives v 5 = y;
And transposing 1 gives 5 + = y;

And therefore x = y +1= 4V 5 + ž.
And if these expressions be turned into numbers, by ex-

tracting the root of 5, &c, they give x = 2.6180+,

and y = 1.6180 +. 5. There are four numbers in arithmetical progression, of

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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, .r+y, x+2y, x+3y, will be the four numbers.
Hence by the 1st condition x2 + 3xy = 22,
And by the 2d 22 - 3.xy + 2y = 40.
Then subtracting the first from the 2d gives 2y = 18;
And dividing by 2 gives y = 9;
And extracting the root gives y

3.
Then substit. 3 for y in the 1st, gives ta 4-9.r = 22;
And completing the square gives x + 9.r +-=;
Then extracting the root gives x +4=;
And transposing gives r = 2 the least number:
Hence the four numbers are 2, 5, 8, 11.'

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

Let x, y, and > denote the three numbers sought.
Then by the 1st condition rz =y",
And by the 2d x +y +z =7,
And by the 3d r? + y + z = 21.
Transposing y in the 2d gives x + z = 7-Y;
Sq. this equa. gives .22 + 2x2 + 2* = 49 - 144 +yo;
Substi. 2y2 for 2.xz, gives x2+2y? +z+=49- 14y+y;
Subtr. y' from each side, leaves ? + y +=49-14y;
Putting the two values of x2 + y2 + 22

21=49- 14y;
equal to each other, givess
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 y + z = 5;
Transposing z in the last, gives x = 5-z;
This substit. in the next above, gives 52-z* = 4;
Changing all the signs, gives 24 – 5% = -4;
Then completing the square, gives z’ -- 52 + 2 = 1;
And extracting the root gives %- { = +1;
Then transposing t, 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 42?

Ans. 6.

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