Solution of Binomial Equations.

SECTION V.

Binomial Equations.

223. Definition. When an equation with one unknown quantity is reduced to a series of monomials, and all its terms which contain the unknown quantity are multiplied by the same power of the unknown quantity, it may be represented by the general form

AxM0,

and may be called a binomial equation.

224. Problem.

To solve a binomial equation.

Solution. Suppose the given equation to be

Hence, find the value of the power of the unknown quantity which is contained in the given equation, precisely as if this power were itself the unknown quantity, and the given equations were of the first degree. Extract that root of the result which is denoted by the index of the power.

225. Corollary. Equations containing two or more unknown quantities will often, by elimination, conduct to binomial equations.

Examples of Binomial Equations.

226. EXAMPLES.

1. Solve the two equations

xy+2y7-4 y3-8x+160,

x2 y7 — 4 y7 — 4 x y3 + 8 y3 + 32 x — 64 — 0.

Solution. The elimination of y between these two equations, by the process of art. 155, gives

being substituted in the first of the given equations, pro

as will be shown when we treat of the theory of equations

Again, the value of x,

x=- 2,

being substituted in the first of the given equations, produces

whence we have

· 4 y3 + 32 = 0,

y3 = 8,

y = 2 or =

as will be shown in the theory of equations.

Examples of Binomial Equations.

10. What number is it, whose half multiplied by its third part, gives 864 ? Ans. 72.

11. What number is it, whose 7th and 8th parts multiplied together, and the product divided by 3, gives the quotient 298 ? Ans. 224.

12. Find a number such, that if we first add to it 94, then subtract it from 94, and multiply the sum thus obtained by the difference, the product is 8512. Ans 18.

13. Find a number such, that if we first add it to a, then subtract it from a, and multiply the sum by the difference, the product is b. Ans. ✔ (a2 —b).

14. Find a number such, that if we first add it to a, then subtract a from it, and multiply the sum by the difference, the product is b. Ans. (a+b).

15. What two numbers are they whose product is 750, and quotient 3? Ans. 50 and 15. 16. What two numbers are they whose product is a, and quotient b?

Ans. ab and ✔

a

b

17. What two numbers are they, the sum of whose squares is 13001, and the difference of whose squares is 1449? Ans. 85 and 76.

18. What two numbers are they, the sum of whose squares is a, and the difference of whose squares is b? Ans. (a+b) and (ab).

19. What two numbers are to one another as 3 to 4, the sum of whose squares is 324900 ? Ans. 342 and 456.

20. What two numbers are as m to n, the sum of whose

squares is a ?

Ans.

ma

and ✓ m2 + n2)

n √ a 、 (m2+ n2)°

Examples of Binomial Equations.

21. What two numbers are as m to n, the difference of

whose squares is a?

Ans.

and

n√ a

√ (m2 — n2) (m2 — n2)° 22. A certain capital is let at 4 per cent.; if we multiply the number of dollars in the capital, by the number of dollars in the interest for 5 months, we obtain 1170413. What is the capital? Ans. $2650.

23. A person has three kinds of goods, which together cost $5525. The pound of each article costs as many dollars as there are pounds of that article; but he has one third more of the second kind than he has of the first, and 3 times as much of the third as he has of the second How many pounds has he of each?

Ans. 15 pounds of the first, 20 of the second, and 70 of the third.

24. Find three numbers such, that the product of the first and second is 6, that of the first and third is 10, and the sum of the squares of the second and third is 34.

25. Find three numbers such, that the product of the first and second is a, that of the first and third is b, and that of the second and third is c.

26. What number is it, whose third part, multiplied by

ts square, gives 1944?

Ans. 18.

27. What number is it, whose half, third, and fourth, multiplied together, and the product increased by 32, gives 4640? Ans. 48.

23. What number is that, whose fourth power divided by 1th of it, and 167 subtracted from the quotient, gives the remainder 12000? Ans. 11.