24. 4x3 — 3 (x1+1) (x21 — 2) = x21 (10 − 3 x31). 25. (x3 — 2) (x* −4) = x3 (x} − 1)2 — 12. = — 28. x2+2a2x2= 3 a. 26. x3 — 4x3 — 96. 81 27. x+x-1= 2.9. 29. 81 Vx+ 52x. 3/ 290. Equations may be solved with respect to an expression in the same manner as with respect to a letter. (1) Solve (2x) — 8 (x2- x)+12=0. (2) Solve 5x-7x2 — 8√7x2 — 5 x + 1 = 8. Change the signs and annex + 1 to both sides. 7x2-5x+1+8√7x2 − 5 x + 1 - 7. --- Solve with respect to √7x2-5x+1. (7x2 −5x+1)+8 (7x2 − 5 x + 1)2 + 16 = 9. (7x2 -5x+1)+4 = ± 3. NOTE. In verifying the values of x in the original equation, it is seen that the value of √7x2 -5x+1 is negative. Thus, by putting O for x the equation becomes 0-8√I=8; and by taking - 1 for VI we have (-8)(-1) = 8; that is, 8 = 8. == 291. An equation like that of (3) which will remain un 1 altered when is substituted for x, is called a reciprocal equation. х It will be found that every reciprocal equation of odd degree will be divisible by x − 1 or x + 1 according as the last term is negative or positive; and every reciprocal equation of even degree with its last term negative will be divisible by 21. In every case the equation resulting from the division will be reciprocal. (4) Solve + 2x1 — 3x3-3x2+2x+1=0. This is a reciprocal equation, for, if x-1 be put for x, the equation becomes x-5+2x-4-3x-3-3x2+2x-1+1=0, which multiplied by a gives 1+2x-3x2-3x3 + 2x2 + x3 O, the same as the original equation. The equation may be written (2 + 1) which is obviously divisible by x+1. = + 2 x (îñ3 + 1) − 3 x2 (x + 1) = 0, The result from dividing by x + 1 is xa + x3 − 4 x2 + x + 1 = 0, or (x2 + 1) + x (x2 + 1) = 4a2. By adding 2x2 to (x+1) it becomes (x4 + 2x2 + 1) = (x2 + 1)2. Therefore, including the root - 1 obtained from the factor x + 1, the five roots are −1, 1, 1, § (− 3 ± √5). By this process a reciprocal cubic equation may be reduced to a quadratic, and one of the fifth or sixth degree to a biquadratic, the solution of which may be easily effected. 5. 3(2 a2 — x) — (2 x2 — x)2 = 2. 6. 15x-3x2+4(x2-5x+5)=16. 7. x2+x2+x+x¬1= 4. 9. x2+x+¿ (x2+x) * = Z. 11. x-1=2+2x ̄. 12. √3x+5-√3x-5= 4. 13. (x + 1) − x (x2 + 1) = − 2x2. 15. x2-4x√x+2=12x2. 16. √2x+a+ √2x-a=b. 17. √9x2+21x+1−√9x2+6x+1=3x. 18. x-4x+x ̃ ̄ +4 x ̄ } : = −7. 30. 2(x13 — 1)−1 — 2 (x31 — 4)−1 — 3 (x2 − 2)-1. CHAPTER XIX. LOGARITHMS. 292. In the common system of notation the expression of numbers is founded on their relation to ten. Thus, 3854 indicates that this number contains 103 three times, 102 eight times, 10 five times, and four units. 293. In this system a number is represented by a series. of different powers of 10, the exponent of each power being integral. But, by employing fractional exponents, any number may be represented (approximately) as a single power of 10. 294. When numbers are referred in this way to 10, the exponents of the powers corresponding to them are called their logarithms to the base 10. For brevity the word "logarithm" is written log. |