1. Clearing of fractions, 2x+1+2√2x2+x=21 2. Placing 2 under the rad. sign, 2x+1+√8x2+4x=21 3. Transposing and uniting, ✔8x2+4x=20-2x 4. Involving, 8x2+4x=400-80x+4x2 5. Transposing and uniting, 4x2+84x=400 6. Dividing by 4, x2 +21x=100 7. Completing the square, (Art. 311.b.) 4x2 +84x+441= 400+441 8. Extracting the sq. root, 2x+21=29, 2x=8, and x=4. Example 24. Reduce 2√x−a +3√2x= 7a+5x √x-a. 1. Clearing of fractions, 2x-2a+3√2x2-2ax=7a+5x 2. Transposing and uniting, -3x+3√2x2-2ax=9a 3. Dividing by 3, −x+√2x2 −2ax=3a 4. Transposing and involving, 2x2 −2ax=x2+6ax+9a2 5. Transposing and uniting, x2-8ax+16a2=9a2+16a2 6. Extracting the square root, x-4a=5′′, and x= =9a. Example 25. Reduce x+16-7√x+16=10−4√x+16. 1. Transposing and uniting, x+6=3√x+16 2. Involving, x2+12x+36=9x+144 3. Transposing and uniting, x2+3x=108 4. Completing the square, (Art. 311. b.) 4x2+12x+9= 432+9 5. Extracting the square root, 2x+3=21, and x=9. Example 26. Reduce √x5+√x3=6√x. 1. Divide by √x, x2+x=6 2. Completing the square, (Art. 311. b.) 4x2+4x+1= 24+1 3. Extracting the square root, 2x+1=5, and x=2. 1. C. F. 156x3 +169x2-455x-39x3 +91x2=27x3+ (132x2+161x 2. Divide by x, 156x2+169x-455-39x2+91x=27x2 + 132x+161 3. Transpose and unite. 90x2 +128x=616 4. Divide by 2, 45x2+64x=308 5. Completing the square, (Art. 311. b.) 8100x +11220x 5 +4096=55440+4096 6. Extracting the square root, 90x+64=244, and x=2. 1. Clearing of fractions, 15x3 +30x2+180x2 - 30x3 — 66x3-11x4+132x2-22x3 = 2. Divide by x2, 15x+30+180-30x=66x-11x2+ 132-22x 3. Transposing and uniting, 11x2-59x=-78 4. Completing the square, (Art 311. b.) 484x2-2956.x +3481-3432+3481 5. Extracting the square root, 22x-59-7, and x=3. Example 29. Reduce(x—5)3 — 3(x—5)3⁄43 —40. 1. Reduce to the same index (x-5)-3(x-5) —40 3. Extracting the square root, 2(x-5)—3—13 Example 30. Reduce x+x+6=2+3√x+6. 1. Transposing and uniting, x+2=2√x+6 2. Involving, x2-4x+4=4x+24 3. Transposing and uniting, x2 -8x=20 4. Completing the square, x2-8x+16=36 5. Extracting the square root, x-4=6, and x=10. PROBLEMS PRODUCING QUADRATIC EQUATIONS. I. Let x=the y'ds silk; then 110-x-the y'ds cotton. 1. By supposition, 400-80X(110-x)-x2 2. By expanding, x=-40+/10000=-40+100 3. By uniting a 60, or 140; but as -140 is negative it is not applicable to goods which a man has in his possession; therefore x=60=the yards of silk, 110-x=110—60 =50=the yards of cotton. II. Let x, and 45-x-the ages of the two brothers; then, (45-x)Xx=500 1. Expanding and changing signs, x2-45x=-500 2. Completing the square, (Art. 311. b.) 4x2-180x+ 1825 2000+1825 3. Extracting the square root, 2x-45——5 4. Transposing and uniting, x=20=age of the younger, and 25 age of elder. III. Let x=one number, and x+4=the other; then, 1. By the conditions, (x+4)×x=117 2. By expanding, x2+4x=117 3. By completing the square, x2+4x+4=4+117 4. By extracting the square root, x+2=±11 5. By transposing and uniting, 9, and 13 the numbers. IV. Let x=the gain; then 30+x=what the cloth was (30+x)×x=30x+x2=x3 sold for. 1. Dividing by a and transposing, x2-x=30 2. Completing the square, (Art. 311. b.) 4x2-4x+1= 120+1 3. Extracting the square root, 2x-1=±11 The values of x are +6, and -5, but the last must be rejected, for gain must be considered positive. V. Let x=the less number, then x+3=the greater. 1. By supposition, (x+3)3-x3-117 2. By expand. and transpos. 9x2 +27x=117-27-90 3. Divide by 9, x2+3x=10 4. Complete the square, (Art. 311. b.) 4x2+12x+9 =40+9 5. By evolution, 2x+3=+7, 2 & 5=the numbers. VI. Let x, and x+12=the numbers. 1. By the given conditions, x2+24x+144+x2=1424 2. By transposing and uniting, 2x2+24x=1280 3. Dividing by 2, x2+12x=640 4. Completing the square, (Art. 305.) x2+12x+36= 640+36 5. By extracting the square root, x+6=26 6. Transposing and uniting, 20, and 32=the numbers. VII. Let x=the less, & x+120=the greater prize; then, 1. By the given conditions, +120; x:x:10 2. Conv. the propor. into equation, 10x+1200=x2 3. Transposing and changing signs, x2-10x=1200 4. Completing the sq. (Art. 305.) x2-10x+25=1200+25 5. Extracting the square root, x-5=35, and x=40 Therefore the numbers are 40, and 160. VIII. Let the less; then 6-x=the greater. 1. By the given conditions, x3 +(6—x)3=72 2. Expanding, x3+216-108x+18x2-x3-72 3. Transposing and uniting, 18x2-108x=-144 4. Dividing by 18, x2-6x=-8 5. Completing the square, (Art. 305.) x2-6x+9=−8+9 6. Extracting the square root, x-3=+1 7. Transp. and uniting, x=2=the less, and 4 the greater. IX. Let x=the less number, and 56-x=the greater. 1. By the given conditions, xX(56—x)=640 2. By expanding and changing signs, x2-56x=-640 3. Comp. the sq. (Art. 305.) x2 -56x+784-640+784 4. Extracting the square root, x-28=±√144 +12 5. Tr. and uniting, x= =16=the less, and 40=the greater. X. Let x=the number of pieces; then, 1. By the given conditions, 48x-675 675 2. Clearing of fractions, 48x2—675x=675 3. Dividing by 3, 16x2-225x=225 4. Completing the square, (Art. 311. b.) 1024x2-14400x+ 50625-14400+50625 5. Extracting the square root, 32x-225=255 6. Tr. and uniting, 32x=480, & x=15=the No. of pieces. XI. Let x+3=A's, and r=B's hourly progress; then, 1. By the conditions of the problem, x+3; 150::*: 150-2 25x 2. Conv. prop. into an equation, 375x-25x2+1350. =150x 3. Clearing of fractions, 375x-25x2+1350=450x 6. Completing the sq. (Art. 311. b.) 4x2+12x+9=216+9 XII. Let x, and a +6=the two numbers; then, 1. By the given conditions, 2x2 +47=x2+12x+36 2. Transposing and uniting, x-12x=-11 3. Comp. the sq. (Art. 311. b.) 4x2-48x+144—~44+144 4. Extracting the square root, 2x-12=10, and x=11. The two numbers are 17 and 11. XIII. Let x=the number of persons which B relieved; then x+40=the number which A relieved. 1. By the given conditions, 5x=. 1200 -X 40 x+40 2. Clearing of fractions, 5x2 +200x=48000 3. Dividing by 5, x2 +40x=9600 4. Comp. the sq. (Art. 305.) x2 +40x+400=9600+400 5. Extracting the square root, x+20=100, and x=80. A relieved 120 persons, and B relieved 80. XIV. Let x=the less number, then 10-x=the greater. 1. By the conditions, x2+100-20x+x2=58 2. Transposing and uniting, 2x2-20x=-42 3. Dividing by 2, x2-10x=-21 4. Completing the sq. (Art. 305.) x2-10x+25=-21+25 5. Extracting the square root, x-5=2, and x=7. The numbers are 7 and 3. XV. Let x=the number in the company at first; then, 1. By the conditions of the problem, 175 175 2. Clearing of fractions, 175x=175x-350+10x2 — 20x 3. Transposing and uniting, -10x2+20x=-350 4. Changing signs and dividing by 10, x2-2x=35 5. Completing the square, (Art. 305.) x2-2x+1=35+1 |