1 1 5. Given√x=22, to find the value of x. Ans. x=49 6. Given x+5x+10=8, to find the value of x. Ans. x=3 7. Given 10+x-1/10+x=2, to find the value of x. Ans. x=6 8. Given 2xa —x2+96=99, to find the value of x. 9. Given xo+20x3 —10=59, to find the value of x. Ans. x=3/3 10. Given 3x2n — 2x2+3=11, to find the value of x. Ans. x=1/2 2 1 2 x2, 12. Given √3+2x+2, to find the value of x. Ans. x= 13. Given x√(x--x)=1+** 1 3C to find the value of x. Ans. x=(1+&{√2)* 14. Given -1-x3x2, to find the value of x. 15. Given x√(−1)=√(x2 — b2), to find the value of x. 1 Ans. x=a+√862+a2 16. Given ✓I+x-x2 — 2(1+x=x2)='', to find the 17. Given √(x−1)+√(1−1)=x, to find the value of x. Ans. x√5 18. Given x-2x3n+x=6, to find the value of x. Ans. x=13 QUESTIONS PRODUCING QUADRATIC The methods of expressing the conditions of questions of this kind, and the consequent reduction of them, till they are brought to a quadratic equation, involving only one unknown quantity and its square, are the same as those already given for simple equations. 1. To find two numbers such that their difference shall be 8, and their product 240. Let x equal the least number. Then will x+8=the greater. And x(x+8)=x2+8x=240, by the question, Whence x=-4+16+240=−4+256, by the common rule, before given, Therefore x=16-4=12, the less number, 2. It is required to divide the number 60 into two such parts, that their product shall be 864. Let x=the greater part, Then will 60-x=the less, And x(60-x)=60x-x2=864, by the question, Or by changing the signs on both sides of the equation x2-60x864, Whence x 30 = (900-864) 30/36=30±6, by the rule, And consequently x=30+6=36, or 30-6-24, the two parts sought. 3. It is required to find two numbers such that their sum shall be 10(a), and the sum of their squares 58(b). Let x the greater of the two numbers, Then will a-x=the less, And x2+(ax)2=2x2-2ax+ab, by the question, Or 2x2-2ax=b-a2, by transposition, 10 1 by the rule, And if 10 be put for a, and 58 for b, we shall have *= 116-100-7, the greater number, 10 1 And 10-x- 2-21 ✓116 116-100-3, the less, 4. Having sold a piece of cloth for 241., I gained as much per cent. as it cost me; what was the price of the cloth ? Let x pounds the cloth cost, Then will 24-x- the whole gain, Whence x=-50+✓✓/2500+2400=-50+70=20 And consequently 204 price of the cloth. 5. A person bought a number of sheep for 80l., and if he had bought 4 more for the same money, he would have paid 17. less for each; how many did he buy? Let x represent the number of sheep, And 80x+320=80x+x2+4x, by the same, 6. It is required to find two numbers, such that their sum, product, and difference of their squares, shall be all equal to each other. Let x=the greater number, and y= the less. Then Hence 1= x+y=xy x2 -ya x+y x+y=x3y2 by the question. -=x-y, or x=y+1, by 2d equation. And (y+1)+y=y(y+1) by 1st equation, Whence y=' +√(+1)=1+1 √5, by the rule, 3 1 And xy+1+5=2.6180... Where. I denotes that the decimal does not end. 7. It is required to find four numbers in arithmetical progression, such that the product of the two extremes shall be 45, and the product of the means 77. Let x= least extreme, and y= common difference, Then x, x+y, x+2y, and x+3y, will be the four numbers, Hence tion, (x+3)=x2+3xy=45 { x+y(x+2y)=x2+3xy+2y=77} by the ques And 2y2=77-45-32, by subtraction, Or y2 = 32 16 by division, and y=√16=4, 2 Therefore x2+3xy=x2+12x=45, by the 1st equation, And consequently x-6+ (36 +45) = 6+9=3, by the rule, - Whence the numbers are 3, 7, 11, and 15. 8. It is required to find three numbers in geometrical progression, such that their sum shall be 14, and the sum of their squares 84. Let x, y, and z be the three numbers, Then xz=y2, by the nature of proportion, x+y+z=14 And x2+y+22=84 by the question, Hence x+2=14-y, by the second equation, And x2+22x+22=196-28y+y, by squaring both sides, Or x2+z2+2y2=196-28y+y2 by putting 2y2 for That is x2+y2+22=196-28y by subtraction, Hence y= 196-84 28 =4, by transposition and division, Again xzy2=16, or x= 16 by the 1st equation, 2 , |