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76. DIVISION.

Although in multiplying two imaginary quantities ✔—a and √―b, the result which arises from replacing these quantities by ✔a and b must have its sign changed; yet in division the substitution may be made without any correction being necessary in the sign of the result; for

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the same results as we should get if in each case the minus under the radical were changed into plus; so that the following is the rule for division:

RULE.-Divide the one imaginary by the other, as if the sign under each radical were plus instead of minus, and the result will be correct. But in general, when the divisor is not monomial, it will be better to convert the operation into multiplication, by introducing into dividend and divisor such a factor as will render the latter rational (page 156).

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(1) Multiply

EXAMPLES FOR EXERCISE.

3-5 by --2√-6.

(2) Multiply 2-3 by -7.
(3) Multiply 2+3-2 by 3-2-1.
(4) Multiply a+b-1 by c+d-1.
(5) Divide 3-7 by 4-3.
(6) Divide 2-5 by -3-3.
(7) Divide 2+V-2 by 3--3.
(8) Divide 4+V-2 by 2-√3.
(9) Divide 3+2-1 by 3-2-1.

(10) Divide a+b√−1 by c+d√—1. (11) Find the square of a+b-1.

(12) Find the square root of a + b√ — 1.

NOTE. This last example may be solved by aid of the formula at page 159: for in that, as well as in the theorems at page 158, the imaginary may be used instead of the ordinary surd forms there considered. It may be noticed here that the results of examples (4) and (10) will be of the same form as the original expressions, that is of the form p+q√−1; so that however many expressions of this kind be combined by multiplication and division, the results will always be of the same common form: the form also recurs in the result of example (12); and it may be proved generally, that any power or root, positive or negative, of a+b√—1, is always of the same form as the original. (See the " General Theory of Equations," pp. 28, 34.) It has already been noticed (page 44) that the difference of two squares x2-ye may be conveniently decomposed into two real factors, namely, x+y and a-y: the sum of two squares x2+y2 may be similarly decomposed into two factors, but here the factors will be imaginary, namely, x+y√—1 and r—y√—1, as is evident. may be remarked, finally, that every imaginary form, such as p+—r, may be made to involve only the simple imaginary factor -1; for if we put Vrq, the form becomes p+q-1: so that this may be considered as the general type of all the imaginary square-root forms of common algebra, and, in employing these forms in analytical investigations, this is the type with which they are usually made to agree at the outset.

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CHAPTER VII.

QUADRATIC EQUATIONS.

INVOLVING ONLY ONE UNKNOWN QUANTITY.

77. A quadratic equation, or an equation of the second degree, is an equation into which the second, but no higher power of the unknown quantity enters; and it is, moreover, such, that this second power can be removed only by the extraction of the square root.

If an equation present itself in which the square of the unknown quantity may be removed in any other way: that is to say, by means of any transformations or changes, not requiring this process of evolution, such an equation would not be considered as a quadratic; for the square being removable without extraction, it would be considered only as a simple equation disguised as a quadratic; instances of such equations have been already given (pp. 111, 112, &c.)

The general form of a quadratic equation, when the unknown terms are all brought to one side of the sign of equality and the known terms to the other, is ax2+ bx=c, or else the still more

simple form ax=c; when the equation is reducible to this latter form, in which x in the first power is absent, the equation is called a pure quadratic; but when the first and second powers both enter, the equation is called an adfected quadratic.

To any one competent to solve a simple equation, the solution of a pure quadratic can offer no difficulty; for since, in the latter, a2 is the only unknown quantity that enters, its value is just as easily found as if it were x instead of x2; thus, in the general form, ax2=c, x=; but instead of this being the final

C

a

step, as it would be if we were dealing with a instead of x2, another step is necessary, namely, the step x. This is

a

the only particular in which the solution of a pure quadratic differs from the solution of a simple equation: there is one additional step, the extraction of the square root, a step so obviously suggested by the immediately preceding one, that there is no room for any additional instruction as to the course to be taken. On this account we have not scrupled to introduce such equations under the head of simple equations, so that, in what follows, adfected quadratic equations exclusively will form the subject of examination; because it is only in reference to this class of quadratics that the learner can feel the want of any further instructions, or require the aid of any new principle.

It may be proper, however, here to recall to his remembrance the fact, that the square root of a quantity is always entitled to the double sign in the absence of distinct stipulation to the contrary: a quadratic equation, therefore, whether pure or adfected, is thus said to have two roots. In extracting the square root of the unknown side of an equation, however, this double sign is not to be introduced: the known side of an equation is the complete interpretation of the unknown, as well in reference to sign as to value; we have no more right to give a sign to an unknown expression, than we have to assume for it a value; the symbol of an unknown quantity embodies in itself both the sign and the value of the quantity it symbolizes; and it is to the known side alone that we are to look for every information respecting its interpretation. If in the case of the pure quadratic above, the square root of £ have the value r, then the so

a

lution would be x=r; to introduce any sign, in connexion

with the x, would be to assume that we know something about this unknown quantity, irrespective of its interpretation on the right hand side of the equation.

We make these preliminary remarks, because some writers on algebra have contended that the unknown side of a quadratic equation, after extraction, is as much entitled to the double sign as the known side, forgetting that the unknown root is entitled to nothing but what the known side, which is its interpretation, authorises.

78. It has already been seen (page 71.) that every quadratic expression, that is, every expression of the second degree, may be decomposed into two factors of the first degree, which factors are equal to each other only when the expression is a complete square. Thus taking the general quadratic form a2+ px+q, and extracting the square root, we have

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so that the two factors of x2+px+q are

x+1p+ √(1 p2-q)

x+} p−√ (} p2—q)

which multiplied together reproduce the proposed form.

Now if all the terms of a quadratic equation be brought to one side of the sign of equality, the other side will be zero or 0; and if all be divided by the coefficient of x2, the resulting form will be that above equated to zero, namely, x2+px+7=0, which as we have just seen is the same as

{x+1p+ √({p2 —q)}{x+\p−√ (} p2—q)}=0... (A). And it is plain that this equation will be satisfied provided we satisfy either of the simple equations

x + ≤ p + √( { p2 −q)=0, or x+1p--√(} p2—q)=0...(B); because, to render the product of two factors zero, it is evidently sufficient that one of them be rendered zero; and it is equally obvious that no value of a can render the product (A) zero except a value which renders one of its factors zero.

It thus appears that the complete solution of the quadratic

equation x2+px+9=0, is effected by solving each of the two simple equations (B); that is to say, the two roots of the quadratic are

x=—}p—√ (} p2-q), and x--1p + √ ({p2—q)...(C). This reduction of quadratic to simple equations is what the learner must have been prepared to expect from his observation of what has preceded: equations with two unknowns were reduced to those with one; equations with three unknowns to those with two; and now, in like manner, equations of the second degree, to those of the first.

The formulæ of solution (C) may be arrived at in another manner: we know that the square of a binomial x + a consists of three terms x2+2ax+a2, in which expression we see that the third term, a', is equal to half the coefficient, 2a, of the second term squared. Hence, if the third term of the square be missing, or suppressed, we can at once recover it by aid of the coefficient of the second term: we shall merely have to take half the coefficient of the second term and square it. Thus, suppose the two terms x2+6x were given to complete the square, then the half of the coefficient of the second term being 3, and the square of this 9, we know that the complete square must be x2+6x+9, and that the root of it is x+3; x being the root of the first term, and 3 that of the last, or half the coefficient of the second. Again, suppose the two terms x2-8x were given to complete the square; then, taking half the coefficient of x, we have -4, the square of which is 16; hence the complete square is x2-8x+16, of which the root is x, together with half the coefficient of x, namely, x-4. And in this way may the square always be completed when the first and second terms of it are furnished; so that generally x2+px being proposed, it may be made a square by adding (p)2=p2 to it. Taking, therefore, the equation x2+px+9=0, above, we have, by transposing, x2+px=-q, and adding p2 to each side, that is, completing the square, we have x2+px+\p2= p-q extracting the root, x+1p=√({p2-q); and therefore, considering the result of the extraction as plus or minus, we have

x=— { p+√({p2 —q), or x=-1p-√({p2—q), the same formulæ of solution as before obtained. Any quadratic equation, therefore, when once reduced to the form x2+

+q=0, or to the form x2+px=-q, may be readily solved

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