2 Then Vx+4x+4=144 by extracting the root; Or x+2=12, And therefore x= 12-2=10. 2. Given x2-6x+8=80; to find r. First, x2-6x=8C-8=72 by transposition; 3. Given 2x+8x-20=70; to find x. And x2+4x+4=49 by completing the square; 4. Given Sx-3x+6=5; to find x. Here x2-x+2=17 by dividing by 3, Also x3-x+1=13—2+1=3 by completing the square, Therefore x=+=}• x2 5. Given +20=42} ; to find x. I 2 3 7. Given ax3-bx+c=d; to find x. Here, ax-bx=d-c by transposition, b dby division; And x2. x= a a 唔 a 4a2 8. Given x*+2ax3-b; to find x. Here, x*+2ax3 +a3=b+a3 by completing the square, And x2+ab+ by evolution; Whence x2=b+a—a, And consequently, x=√ : √b+a—a• 9. Given ax®- -bx2-c=—d ; to find x. First, ax-bx-c-d by transposition, EXAMPLES FOR PRACTICE. 1. Given x-8x+10=19; to find x. 2. Given x2-x-40=170; to find x. 3. Given 3x+2x-9=76; to find x. 4. Given Ans. x=9. Ans. x=15. x +7=20; to find x. 2 3 10. Given 3x”—2x3—— = ; to find x. Ans. x= 27 QUESTIONS PRODUCING QUADRATIC 1. To find two numbers, whose difference is 8 and pro duct 240. Let equal the less number, Then will x+8 equal the greater, And xx x+8=x+8x=240 by the question; Whence x+8x+16=240+16=256 by completing the square; Also x+4=256-16 by evolution; And therefore x=16-4=12= less number, and 12+8=20 greater. 2. To divide the number 60 into two such parts, that their product may be 864. Let x= greater part, Then will 60-x-less, And xx60x60x-x3-864 by the question, That is, x2-60x——864 ; Whence 2-60x+-900-864+900-36 by completing the square; Also x-3036-6 by extracting the root; And 60-x-6-36=24-less. 3. Given the sum of two numbers =10 (a), and the sum of their squares =58 (b); to find those numbers. Let x greater of those numbers, Then will a-x=less ; Vol. I. 2 And x2+a-x=2x2+a3—2ax=b by the question, Hence these two theorems, being put into numbers, give 7 and 3 for the numbers required. 4. Sold a piece of cloth for 241. and gained as much per cent. as the cloth cost me; what was the price of the cloth? Let x= pounds the cloth cost, Then 24-x= the whole gain; But 100 x x: 24-x by the question, Or x2=100x 24-x=2400-100x ; Whence That is, x+100x=2400; the square, +100x+2500=2400+2500=4900 by completing And x+50-4500-70 by extracting the roots, Consequently, x=70-50-20= price of the cloth. 5. A person bought a number of oxen for 801. and if he had bought 4 more for the same money, he would have paid 11. less for each; how many did he buy? |