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such parts, that, if the first be multiplied by 2, the second by 3, and the third by 4, the three products shall be all equal.
Ans. 4, 31, and 24 26. It is required to divide the number 36 into three such parts, that the first, of the second, and of the į
À third, shall be all equal to each other.
Ans. The parts are 8, 12, and 16 27. A person has two horses, and a saddle, which, of itself, is worth 501. ; now, if the saddle be put on the back of the first horse, it will make his value double that of the second, and if it be put on the back of the second, it will make his value triple that of the first : wbat is the value of each horse ? Ans. One 301. and the other 401.
* 28. If a gives B 58. of his money, B will have twice as much as the other has left ; and if B gives a 5s. of his mo. ney, A will have three times as much as the other has left ; how much had each ?
Ans. A 13s. and B 11s. 29. What two numbers are those whose difference, sum and product, are to each other as the numbers 2, 3, and 5, respectively ?
Ans. 10 and 2 30. A person in play lost a fourth of his money, and then won back 3s., after which he lost a third of what he now had, and then won back 2s. ; lastly, he lost a seventh of what he then had, and after this found he had but 12s. remaining; what had he at first ?
Ans. 203. 31. A hare is 50 leaps before a greyhound, and takes 4 leaps to the greyhound's 3, 'but 2 of the greyhound's leaps are as much as 3 of the hare's how many leaps must the greyhound take to catch the hare ?
Ans. 300 32. It is required to divide the number 90 into four such parts, that if the first part be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product, and quotient, shall be all equal ?
Ans. The parts are 18, 22, 10, and 40
33. The quotient and remainder of a sum in division are, each, 21; and the divisor is 7 less than their sum : what is the pumber to be divided.
Ans. 1050 34. A man and his wife usually drank out a cask of beer in 12 days, hut when the man was from home it lasted the woman 30 days; how many days would the man alone be in drinking it?
Ans. 20 days 35. A general, ranging his army in the form of a solid square, finds he has 284 men to spare, but increasing the side by one man, he wants 25 to fill up the square ; how many soldiers had he?
Ans. 24000 36. If A and B together can perform a piece of work in 8 days, A and c together in 9 days, and B and c in 10 days, how many days will it take each person to perform the the same work alone ?
Ans. A 1414 days, B 1741, and c 23,7
A QUADRATIC EQUATION, as before observed, is that in which the unknown quantity is of two dimensions, or wbich rises to the second power; and is either simple or compound.
A simple quadratic equation, is that which contains only the square, or second power, of the unknown quantity, as
6 ax:=b, or x2 A compound quadratic equation, is that which contains both the first and second power of the unknown quantity; as
6 ax2 +bx=c, or x3 + - =
; where x=va
In which case, it is to be observed, that every equation of this kind, having any real positive root, will fall under one or other of the three following forms :
1. xa tax=b ... where x=
where ==++v6+b). 3. x2-ax=-b.. where x=+ = -b).
2. m - ax=b ...
=+ v Or, if the second and last terms be taken either positively or negatively, as they may happen to be, the general equation
b ar? £bx=+c, or x-x== which comprehends all the three cases above mentioned, may be resolved by means of the following rule :
Transpose all the terms that involve the unknown quantity to one side of the equation, and the known terms to the other ; observing to arrange them so, that the term which contains the square of the unknown quantity may be positive, and stand first in the equation.
Then, if this square has any coefficient prefixed to it, let all the rest of the terms be divided by it, and the equation will be brought to one of the three forms abovementioned.
In which case, the value of the unknown quantity is always equal to half the coefficient, or multiplier of x, in the second term of the equation, taken with a contrary sign, together with the square root of the square of this number and the known quantity that forms the absolute, or third term, of the equation. (c)
(c) This rule, which is more commodious in its practical appli. cation, than that usually given, is founded upon the same princi. ple; being derived from the well known property, that in any quadratic 72 + ax=+b, if the square of half the coefficient a
Note. All equations, which have the index of the unknowo quantity, in one of their terms, just double that of the other, are resolved like quadratics, by first finding the value of the square root of the first term, according to the method used in the above rule, and then taking such a root, or power of the result, as is denoted by the reduced index of the unknown quantity.
Thus, if there be taken any general equation of this kind, as
32m+ ax=b, we shall have, by taking the square root of x2m, and observing the latter part of the rule,
=-EV+), and a=
of the second term of the equation be added to each of its sides, so as to render it of the form
that side which contains the unknown quantity will then be a complete square ; and, consequently, by extracting the root of each side, we shall have
x+a=tvia2 tb, or x=tatvia2 tb,
which is the same as the rule, taking a and b in + or - as they may happen to be,
It may here, also, be observed, that the ambiguous sign +, which denotes both + and -, is prefixed to the radical part of the value of x in every expression of this kind, because the square root of any positive quantity, as as, is either taor-a; for(+a) X (ta), or (-a)x(-a) are each - taz; but the square root of a negative quantity, as - as, is imaginary, or una
anassignable, there being no quantity, either positive or negative, that, when multiplied by itself, will give a negative product.
To this we may also further add, that from the constant occur. rence of the double sign before the radical part of the above expression, it necessarily follows, that every quadratic equation mus, have two roots ; which are either both real, or both imaginary, according to the nature of the question.
And if the equation, which is to be resolved, be of the following form
am – we shall necessarily have, according to the same principle,
1. Given x2 +4x=140, to find the value of x.
Here x3 +4x=140, by the question,
Whence x=-2+74+140, by the rule,
Where one of the values of x is positive and the other negative. 2. Given x2 12x+30=3, to find the value of x.
Here x2 12x=3.-30=-27, by transposition,
Where it appears that w has two positive values. 3. Given 282 +82 – 20=70, to find the value of a.
Here 2.02 +8x=70+20=90, by transposition,
Therefore x=-2+7=5, or =- -2-7=-9,