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It is required to divide a given number a into two such parts, that if r times one part be added to s times the other part, the sum will be a given number b.
6 · Ans. The part to be multiplied by r is
and the part to be multiplied by s is
In what cases will one or both of these results be negative ? Can both be negative at the same time ?
How are the negative results to be interpreted ? In what cases will either of them become zero ? Can both become zero at the same time? What is to be understood when one or both become zero? In what cases, will one or both become infinite or impossible ? Can either of them ever be of the form?
XXVII. Equations of the Second Degree.
1. A boy being asked how many chickens he had, answered, that if the number were multiplied by four times itself, the product would be 256. How many had he ?
Let x = the number,
4 x x x = 4 x? By the conditions 4 x2 256
: 64 That is
X X = 64. This equation is essentially different from any which we have hitherto seen.
It is called an equation of the second degree, because it contains x*, or the second power of the unknown quantity. In order find the value of x, it is necessary to find what number, multiplied by itself, will produce 64. We know immediately by the table of Pythagoras that 8 X 8 = 64. Therefore
Ans. 8 chickens. Note. The results of these equations may be proved like those of the first degree.
X = 8.
Ans. 7 years.
2. A boy being asked his age, answered, that if it were multiplied by itself, and from the product 37 were subtracted, and the remainder multiplied by his age, the product would be 12 times his age. What was his age ?
x x x= (x – 37) x = — 37 r. By the conditions
zie 37 x 12 x. Dividing by die
x 37 = 12
x2 = 49
X = 7. 3. There are two numbers in the proportion of 5 to 4, and the difference of whose second powers. is 9. What are the numbers ?
Let x = the larger number,
4x 16 x The second power of
16 z By the conditions 2
25 4. There are two numbers whose sum is to the less in the proportion of 15 to 4, and whose sum multiplied by the less produces 135. What are the numbers ?
Let x = the less, and y = the greater.
Then x + y
4. and x (x + y) = 135.
135 The second gives y = Putting this value of y into the first, it becomes 135
15 x +
Hence it appears, that when an example involves the second power of the unknown quantity, the value of the second
power must first be found in the same manner as the unknown quantity is found in simple equations ; and from the value of the second power, the value of the first power is derived.
It is easy to find the second power of any quantity, when the first power is known, because it is done by multiplication ; but it is not so easy to find the first power from the second. It cannot be done by division, because there is no divisor given. When the number is the second power of a small number, the first power is easily found by trial, as in the above examples. When the number is large, it is still found by trial; but a rule may be very easily found, by which the number of trials will be reduced to very few. The first power is called the root of the second power, and when it is required to find the first power from the second, the process is called extracting the root.
It has been shown, Art. XXIV. that the second power of every quantity, whether positive or negative, is necessarily positive ; thus 3 X 3 = + 9, and also - 3 X-33t 9. So a X a = a’, and also - а Х ca”. Hence
Hence every second power, properly speaking, has two roots, the one positive and the other negative. The conditions of the question will generally show which is the true answer.
XXVIII. Extraction of the Second Root. In order to find a rule for extracting the root, or finding the first power from the second, it will be necessary, first, to observe how the second power is formed from the first.
Let a = 20 and b =7; then a + b = 27.
(a + b)(a + b) = a + 2 ab + 6?.
a = 20 x 20 = 400
7 X 7= 49
The product is formed in precisely the same manner in the usual mode of multiplication, as may be seen, if the products are written down as they are formed, without carrying.
49 140 140 400
Here we observe, 7 times 7 is 49, 7 times 20 is 140, 20 times 7 is 140, and lastly 20 times 20 is 400. These added together make 729, which is the second power of 27.
1st. When the root or first power consists of two figures, the second power consists of the second power of the tens, plus the product of twice the tens by the units, plus the second power of the units.
2d. The second power of 9, the largest number consisting of one figure, is 81 ; and the second power of 10, the smallest number consisting of two places, is 100; and the second power of 100, the smallest number consisting of three places, is 10000. Hence, when the root consists of one figure, the second power cannot exceed two figures ; and when the root consists of two figures, the second power consists of not less than three figures, nor more than four figures.
From these remarks it appears, that we must first endeavour to find the second power of the tens, and that it will be found among the hundreds and thousands.
Let it be uired to find the root of 729. This number contains hundreds, therefore the root will contain tens. cond power of the tens is contained in the 700. 20 x 20 is 400, and 30 X 30 is 900. 400 is the greatest second power of tens contained in 700. The root of 400 is 20. Subtract 400 from 729, and the remainder is 329. This must contain 2 a b + b*, that is, the product of twice the tens by the units, plus the second power of the units. If it contained exactly the
product 2 a b of twice the tens by the units, the units of the root would be found by dividing 329 by twice 20, or 40; for 2 a b divided by 2 a gives b. As it is, if we divide by twice 20 or 40, we shall obtain a quotient either exact, or too large by 1 or 2. 40 is contained in 329, 8 times. Write 8 in the root and raise the whole to the second power. 28 X 28 = 784, which is larger than 729. Next try 7 in the place of 8. 27 X 27 = 729. Therefore 7 is right, and 27 is the root required. The operation may stand as follows.
729 (20 + 7 = 27 root.
329 (40 divisor. 27 x 27 = 729.
What is the root of 1849 ?
18,49 (40 + 3 = 43 root.
249 (80 divisor. 43 X 43 = 1849.
In this example, the second power of the tens will be found in the 1800. 30 x 30 = 900 ; 40 X 40 = 1600 ; 50 X 50 = 2500. The greatest second power in 1800 is 1600, the root of which is 40. Write 40 in the place of a quotient. Subtract 1600 from 1829. The remainder is 249, which divided by twice 40, or 80, gives 3. Add 3 to the root, and raise the whole to the second
power. 43 X 43 = 1849. Therefore 43 is the root required.
It is evident that the result will not be affected, if instead of writing 40 in the root at first, we omit the zero, and then subtract the second power of 4, viz. 16 from the 18, omitting the two zeros which come under the other period. Then to form the divisor, the 4 may be doubled, and the divisor will be 8 instead of 80, and the dividend must be 24, the right hand figure being rejected.