whence Therefore

y=3 or -7.

x=9 or 49.

Although the square root of 9 is generally ambiguous, and may be either +3 or 3, still, in verifying the preceding values, √ can not be taken equal to -3, because 9 was obtained by multiplying +3 by itself. For a like reason, √x can not be taken equal to +7. A similar remark is applicable to several of the following examples.

263. The same method of solution may often be extended to equations in which any algebraic expression occurs with two exponents, one of which is double the other.

Ex. 4. Solve the equation (x2+x)2 — 26(x2+x)=—120.
Assuming x2+x=y, this equation becomes

We have now the two equations,

x2+x=20, and x2+x=6,

the first of which gives x=-5 or +4, and the second gives X=- -3 or +2. Thus the equation has four roots,

−5, +4, −3, +2,

and any one of these four values will satisfy the given equa

tion.

Ex. 5. Solve the equation √x+12+√x+12=6.

Ex. 6. Solve the equation 2x2+√2x2+1=11.

This equation may be written

2x2+1+√2x2+1=12.

264. Equations of the Fourth Degree.-An equation of the fourth degree may often be reduced to an equation containing the first and second powers of some compound quantity, with known coefficients, in the following manner: Transpose all the terms to the first member; then extract the square root to two terms, and see if the remainder (with or without the absolute term) is a multiple of the root already obtained. Ex. 7. Solve the equation x-12x3+44x2-48x=9009. We may proceed as follows:

x-12x3+44x2-48x-9009-0(x2-6x

Hence the given equation may be expressed as follows: (x2-6x)2+8 (x2-6x)=9009.

Ans. x=13 or —7, or 3±3√−10.

Ex. 8. Solve the equation x2-2x3+x=132.

Ans. x=4 or -3, or 1±†√−43.

Ex. 9. Solve the equation x+4x2=12.

Ans, x=±√2 or ±√6.

Ex. 10. Solve the equation x6-8x3=513.

3

Ans. x 3 or -19.

Ex. 11. Solve the equation x+x=756.

Ans. x=243 or -285.

Ex. 12. Solve the equation ——x3 — — z11⁄2·

Ex. 13. Solve the equation 2x3+3x3=2.

Ans. x=1√2.

Ans. xor -8.

Ex. 14. Solve the equation -√x=22z.

Ans. x= =49 or

Ex. 15. Solve the equation √10+x−√10+x=2.

361 9

Ex. 16. Solve the equation x+20x3-10=59..

Ans. x=√3 or V-23.

Ex. 17. Solve the equation 3x2-2x+3=11.

Ans. x=V2 or V.

Ex. 18. Solve the equation √1+−x2−2 (1+x—x2)=}.

Ans. x=÷÷÷√41 or ±±±√11.

Ex. 19. Solve the equation Vx+√x=20.

Ans. x=256 or 625.

Ex. 20. Solve the equation x-4x3+7x2-6x=18.

Ans. x=3 or -1, or 1±√−5.

Ex. 21. Solve the equation x2+5x+4=5√x2+5x+28. Ans. x=4 or -9, or —§±1√√−51.

Ex. 22. Solve the equation x2+3=2√x2−2x+2+2x.

Ans. x=1.

Ex. 23. Solve the equation (x+√x)*—(x+√x)2=20592.

Ans. x 9 or 16.

Ex. 24. Solve the equation x+√25+x=157.

Ans. x=144 or 171.

Ex. 25. Solve the equation √x-1=x−1.

Ans. x=1 or 2.

265. We have seen that every equation of the second degree has two roots; that is, there are two quantities which, when substituted for x in the original equation, will render the two members identical. In like manner, we shall find that every equation of the third degree has three roots, an equation of the fourth degree has four roots, and, in general, an equation of the mth degree has m roots.

Before determining the degree of an equation, it should be freed from fractions, from negative exponents, and from the radical signs which affect its unknown quantities. Several of the preceding examples are thus found to furnish equations

of the fourth degree, while others furnish equations of the second degree.

The above method of solving the equation x2+px"=q will not always give us all of the roots, and we must have recourse to different processes to discover the remaining roots. The subject will be more fully treated in Chapter XXI.

Problems producing Equations of the Second Degree.

Prob. 1. It is required to find two numbers such that their difference shall be 8 and their product 240.

Then will x+8=the greater.

And by the question, x(x+8)=x2+8x=240.
Therefore

x=12, the less number,

x+8=20, the greater.

Proof. 20-12-8, the first condition.

20×12=240, the second condition.

Prob. 2. The Receiving Reservoir at Yorkville is a rectangle, 60 rods longer than it is broad, and its area is 5500 square rods. Required its length and breadth.

Prob. 3. What two numbers are those whose difference is 2a, and product b?

Ans. a±√a2+b, and −a±√a2+b.

Prob. 4. It is required to divide the number 60 into two such parts that their product shall be 864.

Let x=one of the parts.

Then will 60-x-the other part.

And by the question, x(60-x)=60x-x2-864.

The parts are 36 and 24, Ans.

Prob. 5. In a parcel which contains 52 coins of silver and copper, each silver coin is worth as many cents as there are copper coins, and each copper coin is worth as many cents as there are silver coins, and the whole are worth two dollars. How many are there of each?

.

Prob. 6. What two numbers are those whose sum is 2a and product b? Ans. a+Va2-b and a-Va2-b.

Prob. 7. There is a number consisting of two digits whose sum is 10, and the sum of their squares is 58. Required the number.

Let x=the first digit.

Then will 10-x-the second digit.

And

that is,

x2+(10−x)2=2x2-20x+100=58;

x2-10x=-21,

x2-10x+25=4,

x=5±2=7 or 3.

Hence the number is 73 or 37.

The two values of x are the required digits whose sum is 10. It will be observed that we put x to represent the first digit, whereas we find it may equal the second as well as the first. The reason is, that we have here imposed a condition which does not enter into the equation. If x represent either of the required digits, then 10-x will represent the other, and hence the values of x found by solving the equation should give both digits. Beginners are very apt thus, in the statement of a problem, to impose conditions which do not appear in the equation.

The preceding example, and all others of the same class, may be solved without completing the square. Thus,

Let x represent the half difference of the two digits.

Then, according to the principle on page 89, 5+x will represent the greater of the two digits, and 5-x the less. The square of 5+x is 25+10x+ x2,

5-x25-10x+ x2.

50 +2x2, which, according to

=58.
2x2= 8,

ac2 =
4,

x = ±2.

5+x=7, the greater digit,

5-x=3, the less digit.

Prob. 8. Find two numbers such that the product of their sum and difference may be 5, and the product of the sum of their squares and the difference of their squares may be 65.