EXAMPLES OF QUADRATIC EQUATIONS. Example 1. Reduce 3x2 1. Transposing and uniting, 2. Divide by 3, -9x-4-80. 3x2-9x=84. x2-3x=28. 3. Comp. the square, (Art. 311. b.) 4x2-12x+9=112+9. 4. Extracting the square root, 5. Transposing and uniting, Example 2. Reduce 4x 36- -X 2x-3=+11. x=7, or -4. -46 1. Clearing of fractions, 4x2-36+x=46x. 2. Transposing and uniting, 4x2 −45x=36. 3. Completing the square (Art. 311. b.) 64x2-720x+2025 =576+2025. 4. Extracting the square root, 8x-45 +51. 5. Transposing and uniting, x=12, or —3. 1. Clearing of fractions, 4x2+4x-14+x=14x+14. 2. Transposing and uniting, 4x2-9x=28. 3. Completing the square, (Art. 311, b.) 64x2-144x+81 =448+81. 4. Extracting the square root, 8x-9=±23. 5. Transposing and uniting, x=4, or −7. Example 4. Reduce 5x-3x-3 3x-6 =2x+ X- -3 2 1. Clearing of fractions, 10x2-30x-6x+6=4x2-12x +3x2-6x-9x+18. 2. Transposing, 10x2-4x2-3x2-30x-6x+12x+6x+9x =-6+18. 3. Uniting terms, and dividing each side by 3, x2—3x=4. 4. Completing the sq. (Art. 311, b.) 4x2-12x+9=9+16. 5. Extracting the square root, 2x−3=√25. 6. Transposing and uniting, x=4, or —1. 1. Clearing of fractions, 64x2 -100x+9x2=12x3. 2. Dividing by x, 64x-100+9x=12x2. 3. Transposing and uniting, -12x2+73x=100. 4. Changing signs, 12x2-73x=-100. 5. Completing the square, (Art. 311, b.) 576x2-3504x+ 5329-4800+5329 6. Extracting the square root, 24x-73=±23. 7. Transposing and uniting, x=4, or 2. Example 6. Reduce 3x-4 =10—2—2. +1=10x x-4 2 1. Cl. of fr. 6x÷8+2x-8=20x—80—x2+2x+4x−8. 2. Transposing and uniting, x2-18x=72. 3. Comp. square, (Art. 311. b.) x2-18x+81=-72+81. 4. Extracting the square root, x-9=±3, and x= Example 7. Reduce +4_7-x 3 x-3 4x+7 -1. =12, or 6. 1. C.F.9x2-27x+36x-108-189+27x=12x2 -36x+21x -63-27x+81. 2. Transposing and uniting, -3x2+78x=315. 3. Changing signs and dividing by 3, x2-26x=-105. 4. Comp. square, (Art. 311. b.) 4x2-104x+676=-420 +676. 5. Extracting the square root, 2x-26=√256, and x=21, or 5. Example 8. Reduce. x2-10x2+1 =x- -3. x2-6x+9 1. Clearing of fractions, x3-10x2+1=x3—9x2 +27x—27. 2. Cancelling terms, 2+1=27x-27. 3. Transposing and changing signs, x2+27x=28. 4. Comp. square, (Art. 311. b.) 4x2 +108+729=112+729. 5. Extract the square root, 2x+27=±29. 6. Transposing and uniting, x=1, or -28. 1. Clearing of fractions, 6x+2x+2=3x2+3x. 2. Transposing and uniting, -3x2+5x=-2. 3. Changing signs, 3x2-5x-2. 4. Comp. the sq. (Art. 311. b.) 36x2 −60x+25=24+25. 5. Extracting the square root, 6x-5=√49=7, and x=2. 1. Clearing of fractions, 18x-x2+x-2x+2=6x2-54x+ 12x-108. 2. Transposing and uniting, -7x2 +59x=-110. 3. Changing signs, 7x2 −59x=110. 4. Completing the square, (Art. 311. b.) 196x2-1652.x +3481 3080+3481. 5. Extracting the square root, 14x-59=±81, and x=10. 1. Clearing of fractions, ax2 +a3=2ax. 2. Dividing by a and transposing, x2 — 2x——a2. 3. Completing the square, (Art. 305.) x2-2x+1=1-a2. 4. Extracting the square root, x-1=√1-a2. 5. Transposing, x=11-a2. Example 12. Reduce x4 +ax2=b. 1. Comp. square, (Art.314.) x4 +ax2+ a2 ስ? =b+ 4 4 a + a2 b+ -2 32 23 1 2 16 2. Comp. square, (Art. 314.)x 3. Extracting the square root, x3—1=0. 4. Transposing, x3=4, andx=. Example 14. Reduce 2x3+ 3x3—2. 1. Comp. square, (Art. 311. b.) 16x3+24x3 +9=16+9 2. Extracting the square root, 4x+3=5. 3. Transposing and uniting, 4x3=2. 4. Dividing by 4, and cubing, x=f. Example 15. Reduce x-√x=22}. 1. Clearing of fractions and dividing, 3x-2√x=133. 2. Comp. square, (Art. 311. b.) 36x-24/x+4=1596+4. 3. Extracting the square root, 6x-2=40. 4. Tr. dividing by 6, and squaring, x=7, and x=49. Example 16. Reduce 2x4x2+96=99. 1. Transposing and uniting, 2x4 —x2=3. 2. Comp. square, (Art. 311. b.) 16x4 −8x2+1=24+1. Example 17. Reduce (10+x)*— (10+x)‡—2. 2. Extracting the square root, 2(10+x)¤—1—3 3. Transposing and uniting, 2(10+x) —4 4. Dividing by 2, (10+x)=2 5. Involving to the fourth power, 10+x=16, and x=6. Example 18. Reduce 3x2"-2x”=8. 1. Completing the square, (Art. 311. b.) 36x2"-24x+4= 96+4 2. Extracting the square root, 6x-2=10, and 6x=12 3. Dividing by 6, and evolving, "=2, and x=”√/2. Example 19. Reduce 2(1+x—x2)−√1+x−x2: 1. Divide by 2, (1+x−x2) — √1+x−x2 2 2. Comp. the square, (Art. 311. b.) (1+x-x2) — (1+x—x2), 3. Extracting the square root, ✓1+x¬x2−4=√1⁄2 4. Transposing and uniting, 2 5. Involving both sides and trans. x-x2=-1=-} 6. Changing signs, 7. Completing the square, (Art.305.) x2-x+=+&+= }} 3. Extracting the square root, x-√√41 9. Transposing, Example 20. Reduce 3/x3-a3=x—b. 1. Involving each side to the 3d power, x3-a3x3—3bx2 +3b2x-b3 2. Transposing and uniting, -362x+3bx2-a3-b3 3. Dividing by 3b, -bx+x2 = a3-b3 = 4. Completing the square, (Art. 305.) x2 bx+ 4a3-b3 126 1. Clearing of fractions, 2x+2x=16-x 2. Transposing and uniting, 2x=16-3x 3. Involving, 4x=256–96x+9x2 4. Transposing and uniting, -9x2 +100x=256 5. Changing signs, 9x2-100x=-256 6. Completing the square, (Art. 311. b.) 324x2-3600x+ 10000-9216+10000 7. Extracting the square root, 182—100——28 8. Transposing and uniting, 18x=72, and x=4. 6 3 Example 22. Reduce x3 + x5=756 1. Complete the square, (Art. 311. b.) 4x3+4x3+1= 3024+1 2. Extracting the square root, 2x3+1=55 3. Tr. unite and divide, 23–27 4. By evolution and involution, x=3, and x=243. Example 23. Reduce 2x+1+2√x=· 21 √2x+1 |