CHAPTER XXII THE QUADRATIC FORM. HIGHER EQUATIONS. IRRATIONAL EQUATIONS EQUATIONS IN THE QUADRATIC FORM 326. An equation in the quadratic form is an equation having three terms, two of which contain the unknown number; the exponent of the unknown number in one term being twice the exponent of the unknown number in the other term. Thus: 24-13 x2+36= 0; x6 + 7 x3 = 8; Vx2-4Vx=12. HIGHER EQUATIONS SOLVED BY QUADRATIC METHODS 327. It will be seen at once that many equations in the quadratic form must be of a higher degree than the second. The method of factoring permits the solution of many such equations, and is generally employed in elementary algebra. 328. If quadratic factors result from the application of factoring to higher forms of equations, they may ordinarily be solved by completing the square or by the quadratic formula. Such factors most frequently occur in connection with binomial equations. And, (x+2) (x2 - 2 x + 4) (x − 1) (x2 + x + 1) = 0. Factoring, Expressed with fractional exponents and in the transposed form, we Factoring, From which Squaring, Solving, = x2 – 5x+2V 52 = 10. ( 2+2V – 5 2=8. [ (x2 – 5 x − 2) 3 — 2 ][ (x2 − 5 x − 2) 3 + 4] = 0. (x2-5x-2)=2 and (x2 -5x-2)=-4. 329. For convenience of solution a single letter may be substituted for a compound expression in equations in the quadratic form. Care should be taken that no solution is lost in the final substitutions. Solve: Exercise 111 (See also page 413.) 24. (x+1)2(x+1)= 6. 25. (x+2)2 -(x+2)= 20. 26. (x−3)2+5(x − 3) = 6. 27. 2(x+1)2 - (x + 1)= 6. 28. 3(x-5)2-7(x-5)=-2. 29. (2x+1)2-9(2 x+1)+18=0. 30. √2x+1−√2x+1= 6. 31. √x-5-√x-5=42. 36. = (x+5)+2(x+5)* – 15. 37. 5(x+6)+5(x+6)1 = 60. 38. (4x-3)-2√4x-3=15. 39. √3x+1+(3x+1)=20. 40. (x+12)1 + √x+12=6. 41. (x+1)−1+(x + 1) ✈ − 12 − 0. 42. (x−1)1+5√(x −1)−1—— 6. 43. (x — 5)3 —7√ (x — 5)3 — 8. 44. (x−1)—5√(x−1)−1+4=0. 45. Vx+21=12−√x + 21. 46. 4√(x-1)*—5 √(x − 1)2 — — 1. X 57. (x2+2x-1)+5√x2+2x-1=6. 58. x2+2x- 6 −√ x2 + 2 x − 6 = 6. 65. x22x+6√x2-2x+5=11. 66. x2+5x-2√x2-4x+7=9x-7. PROBLEMS PRODUCING AFFECTED QUADRATIC EQUATIONS 330. In the solution of a problem involving a quadratic equation we retain only the result that satisfies the given conditions. Negative results often fail to meet the conditions even though they satisfy the equation that we obtain. Illustration: A company of boys bought a boat, agreeing to pay $60 for it. Three of the boys failed to pay as agreed, so each of the others paid $1 more than he had promised. How many boys finally paid for the boat? Let x = the number of boys finally paying for the boat. the number of boys at first agreeing to share its cost. the number of dollars paid by each boy. the number of dollars each at first agreed to pay. 60 = x + 3 Then From which That is, 12 boys finally shared in the cost of the boat. Exercise 112 (See also page 413.) 1. Divide 30 into two parts whose product is 216. 2. Divide 27 into two parts whose product is 180. 3. Find two consecutive numbers, the sum of whose squares is 265. 4. Find two consecutive numbers, the sum of whose reciprocals is 4. 5. Find the two parts of 18 whose product is 1 more than 8 times their difference. 6. One number is their difference by 105. of another, and their product exceeds Find the numbers. 7. Find two numbers quotient is 21. whose product is 21, and whose |