4th. It is required to find two numbers whose sum and product are equal?

Ans. If x and y denote the two numbers, then z is arbitrary, and

5th. Required three numbers, such that if the first is multiplied by 7, the second by 9, and the third by 11, the first product shall be 1 less than the second, and 2 greater than the third ?

Ans. 5, 4, 3; or 104, 81, 66; or 203, 158, 129;

6th. A farmer buys 124 head of cattle, viz., pigs, goats, and sheep, for 400 £. Each pig costs 4 £. 10 sh.; each goat 3 £. 3 sh. 4 d.; and each sheep 1 £. 5 sh. How many are there of each kind?

Ans. 17, 99, 8; or 40, 60, 24; or 63, 21, 40.

7th. Required three numbers such that the sum of their products by the numbers 3, 5, 7, respectively, may be 560, and the sum of their products by the numbers 9, 25, 49, respectively, may be 2920 ?

Ans. 15, 82, 15; or 50, 40, 30.

8th. Find a number, N, which, being divided by 11, gives the remainder 3; divided by 19, gives the remainder 5; and divided by 29, gives the remainder 10?

Ans. N=4128+6061t; 4128 being the least number which satisfies the

problem.

SECTION VI.

FORMATION OF THE SQUARE AND EXTRACTION OF THE SQUARE ROOT. CALCULUS OF RADICAL QUANTITIES OF THE SECOND DEGREE.

103. The quantity which, raised to the square, reproduces a given quantity, is termed the square root of that quantity. In arithmetic there are no negative quantities, and the absolute values of numbers are alone considered; a square root can therefore have only a single value. The square root of 4, for example, can be no number but 2. But in algebra, into the calculations of which negative as well as positive quantities enter, the case is otherwise; for it follows from Article 4, that not only (+2)2=4 and (+a)2=a2, but also that (-2)2=4 and (—a)2=a2. Therefore, if it is agreed to denote by a a certain quantity of which the square is a, this square root may be either +√a or-a. The two values are generally expressed thus, +a.

+va and -a a are the only square roots of the quantity a. To establish this proposition, let x2=a. Then the different square roots of a are the values which satisfy the equation x2=a or x2-a=0.

But this product

are equal to zero.

Since a=/a2, x2-
2—a=x2-√√ a2=0.

But √ax √aa (Part I. Art. 257),
and (x-√a) (x+√a)=x2−√ax √a;

••.x2-a=(x−√a) (x+√a),

and.. (x-a) (x+√a)=0.

cannot be equal to zero, unless one or both its factors

From x- ✔a=0 is obtained x=√a,

and from x+a=0 is obtained x=-√a.

Whence, since any value of x which is not either +va ora, cannot render either of the factors x-√a, x+√a, equal to zero, it cannot render x2-a=0; therefore the quantity a has no square roots but +a,√a.

Consequently the square root of an algebraic quantity has two values which are equal and of contrary signs.

The expression ✔a is taken to signify +a. Therefore when a negative or subtractive root is meant, it is proper to write —√ā.

-

104. The quantity placed under the radical sign may be negative, as in the instances -4, √-a2. Now the square of every quantity is positive; for by the rule of signs (+a)×(±a)=+a2. Therefore no quantity can have a negative square, and no negative quantity a square root.

The values of the roots of -4, -a are termed imaginary quantities or imaginary roots. In strictness they are not quantities; yet, being operated upon by the rules of the algebraic calculus, this designation is applied to

them.

To mark the difference between quantities which are positive or negative and expressions such as √-a, the former are termed real quantities. Every imaginary expression of the form √-A can be transformed into a product of two factors, the one real, the other equal to √—1.

For the quantity A, if taken additively, must have two square roots which are real. Let a denote one of these roots, that which is positive; then -A may be represented by ay, provided y is determined in such a manner that (ay)2 may be equal to -A; or, which is the same thing, that a2y2=-A.

But since A is the square of a, A=a2,

therefore Aya2y2,

Ay2=-A,
y2=-1,

and y=√1;

therefore ay+a√-1;

but ay√A.;. √−A=+a√−1.

105. Let abc. represent any product :

.....

(abc....)2=(abc....) × (abc ....)=aabbcc.... =a2b2 c2..

Therefore the square of a product is formed by raising each factor to the

square.

The square of a factor a", which has an exponent n, is (a")2=a"×a"=a2”. Whence, the square of any monomial is obtained by squaring the coefficients and doubling the exponents:

Reciprocally, from the preceding principles it is concluded that the square root of a product is obtained by extracting the square root of all the factors; and that the square root of a monomial is obtained by extracting the square root of the coefficient, and dividing the exponents by 2.

These rules give abc=√a×√b×√ č.

/64a+b2=+8a2b.

106. A quantity placed under the sign is termed a radical quantity, or, simply, a radical.

When all the factors of a monomial or of a product are not squares, the extraction of the square root is first indicated; and the radical is afterwards simplified by placing on the exterior of the radical sign all the factors which are squares.

For example, if it is required to extract the square root of 50a5b3c ;
Since 50a b c 2×25 × aa ×a× b2 × c=25a+b2 × 2ac,

since also 50a3b2c=√25a+b2 × √2ac,

In general, a radical of the second degree is simplified by decomposing the quantity placed under the radical sign into two products, of which the one contains only square factors, and the other, factors which are not squares; and then extracting the square root of the first product, and indicating the extraction of the square root of the second.

Hence, if a denotes the square root of A,

√—A=√ a2 ×−1=a√−1.

107. Let a denote the numerator and b the denominator of any fraction; α a a2

(U)1⁄2 b^ b b2

then, by Article 34, () = { x i = r2·

Whence the square of an algebraic fraction is obtained by squaring the terms of the fraction.

Conversely, the square root of an algebraic fraction is obtained by extracting the square root of the terms of the fraction.

Example. Required the square root of

49a+b6

16cd

108. When the terms of a fraction are not perfect squares, the root of each term may be indicated, and, if possible, simplified; but, usually, the denominator is rendered rational. This is accomplished by introducing into both terms of the fraction the factors which render the denominator a perfect square; then the square root of the denominator is extracted, and the radical remains only in the numerator.

3a+b
50c3

Example. Extract the square root of ?

The number 50 becomes a square if multiplied by 2; and the quantity c3, if multiplied by c.

3a+b 3a+b 2c 6a+be

109. Let a+b+c+d be a polynomial, in which a, b, c, d represent any quantities whatever.

If all the terms, excepting the last, are considered as one term, and the square of the polynomial is formed as if it were a binomial, m+d,

then, since (m+d)2=m2+2md+d2,

in like manner {(a+b+c)+d}2=(a+b+c)2+2(a+b+c)d+d2,

{(a+b)+c}=(a+b)2+2(a+b)c+c2,
(a+b)2=a2+2ab+b2;
..(a+b+c+d)2=a2+2ab+b2+2(a+b)c+c2+2(a+b+c)d+d2.

As this method may be extended to include polynomials composed of any number of terms, it follows, from the result obtained by it, that the square of a polynomial consisting of any number of terms is composed of the square of the first term, plus the double product of the first term by the second term, plus the square of the second term, plus the double product of the sum of the first and second terms by the third term, plus the square of the third, &c. &c.

If the products 2(a+b)c, 2(a+b+c)d, &c.,

or 2ac+2bc, 2ad+2bd+2cd,

are formed, it appears that the square of a polynomial is composed of the

344

squares of all its terms, plus the double product of all its terms, taken two by two.

110. Let it be required to extract the square root of any polynomial expression, P.

The terms of the proposed polynomial can be arranged in such a manner that the exponents of the same letter (r) are diminished from left to right. Let the terms be thus arranged, and suppose

P=A+B+C+

...

Also, represent by a+b+c+.... the square root of P, arranged in the

same manner.

must reproduce P; whence, from the order The square of a+b+c+. observed in arranging the terms, and the principle of Article 12', it follows that among the terms which compose the square of a+b+c+... that term in which has the highest exponent is a2; therefore A is the square of a ; consequently the first term of the root is obtained by extracting the square root of the first term of the given polynomial.

Subtracting the square of the first term of the root from P, and denoting the remainder by R,

R=B+C+

R contains the double product of the first term of the root by the second term, plus the square of the second term of the root, &c.

Now it is evident that the double product of the first term of the root by the second term must contain a with an exponent higher than in the other terms of R.

Therefore B is this double product; therefore the second term of the root is found by dividing the first term of R by the double of the first term of the root.

Two terms, a, b, of the root being found, if b is added to 2a and the sum 2a+(+b) is multiplied by b, the product, (2a+b)b=2ab+b2, is equal to twice the product of the first term of the root by the second term plus the square of the second term.

The product (2a+b)b being subtracted from R, a second remainder, R', is obtained, which contains the double product of the sum of the first and second terms of the root by the third term plus the square of the third term, &c.

The term in of this remainder which has the highest exponent must be the double product of the first term of the root by the third; whence it follows that the third term of the root is obtained by dividing the first term of R' by the double of the first term of the root, that is, by 2a.

Let the third term of the root be c; the terms a, b, c of the root, and R", the remainder left by the subtraction of (2a+2b+c)c from R', afford the means of obtaining the fourth term of the root, in the same manner as a, b, the first and second terms of the root, and R', the second remainder, have led to the discovery of the third term of the root c.

When P is a square all the terms of the root must be obtained by continuing this series of operations, for each division gives one term.

111. If the process for the extraction of the square root of a number is compared with that for the extraction of the square root of an algebraic quantity, the periods of figures of the number being considered the analogues of the terms, arranged according to the descending powers of r in the algebraic quantity, it is found that the principal difference arises from the distinction between arithmetical and algebraic quantities; namely, that in the former the operations are executed, and in the latter only indicated.

Arranging the power, root, and divisors as in extraction of the square root of a number, and proceeding in the manner pointed out by the investigation contained in the last Article, the square root of any algebraic quantity may be found, as in the following example:

Example. Required the square root of 4x++12x3 +5x2−6x+1.

The square root of 4+ (by the rule for monomials) is 2x2; the first term of the square root of P is therefore 2x2.

Subtracting the square of 2x2 from P, the first term of R is 12x3.

The double of the first term of the root is 4x2.

12x+34x2=3x; therefore 3r is the second term of the root. (4x2+3x)3x=12x+9x2= the double product of the first term of the root by the second term plus the square of the second term.

Subtracting 12+9x2 from R, the first term of R' is -4x2.

-4x+4x2 (the double of the first term of the root)=-1; therefore -1 is the third term of the root.

(4x2+6x−1)×(−1)=—4x2-6x+1= the double product of the sum of the first and second terms of the root by the third term plus the square of the third term.

Subtracting -4x2-6x+1 from R', it is found that R"=0; therefore 4x++12x3+5x2-6x+1 is a perfect square, and its square root is 2x2+

3x-1.

112. When a remainder =0 is obtained, it is to be concluded that the proposed polynomial is a square, and its square root complete; for, from the manner in which the calculation is made, each remainder which is successively obtained is the given polynomial, diminished by all the parts which compose the square of those terms of the root which have been found by the part of the calculation already made; therefore when the root is complete the remainder must be equal to zero; and, reciprocally, when a remainder equal to zero is found, it is evident that the terms written in the root compose the exact square root of the polynomial proposed.

On the other hand, when P is not a square, the calculation shows that it is not; for, supposing always that the terms of P are arranged according to the descending powers of x, and observing that at each successive subtraction the first term of P, R, R'.... is destroyed, it must happen that the exponent of x is diminished in the first term of each successive remainder, and by consequence in the terms of the root.

This admitted, let K be the last term of P, that is, the term in which I has the least exponent, and let the square root of K be k. Now it is obvious that if P is a square, k must be the last term of its square root; consequently in the course of the calculation k ought to be found; if it is not the calculation must, by the diminution of the exponents, as above mentioned, give a term of the root of a degree inferior to k, in which case it is evident that P is not a square. The conclusion is the same if the calculation gives the term k with a remainder not equal to zero.

113. The preceding explanations are accommodated to the case in which the terms of P are arranged according to the descending powers of x.

But when the contrary order is adopted they still subsist, it being only necessary to make a slight modification in the test by which it is sought to ascertain whether the polynomial is a square.

First, it is obvious that the same process of calculation gives all the terms of the root, and that when the last remainder is 0, P is a square, and the terms found are those of its square root.