square of a binomial, and consequently px as being twice the product of the first term of the binomial by the second, it is evident that the second term of this binomial must be 2 257. In order, therefore, that the expression x2+px may be rendered a perfect square, we must add to it the square of this second term ; and in order that the equality of the two mem p bers may not be destroyed, we must add the same quantity to the second member of the equation. We shall then have Taking the square root of each member, we have p whence, by transposition, x=- ± √ q + 202 4 Thus the equation has two roots: one corresponding to the plus sign of the radical, and the other to the minus sign. These two roots are 258. Hence, for solving a complete equation of the second degree, we have the following RULE. 1st. Reduce the given equation to the form of x2+px=q. 2d. Add to each member of the equation the square of half the coefficient of the first power of x. 3d. Extract the square root of both members, and the equation will be reduced to one of the first degree, which may be solved in the usual manner. 259. When the equation has been reduced to the form x2+ px=q, its two roots will be equal to half the coefficient of the sec ond term, taken with a contrary sign, plus or minus the square root of the second member, increased by the square of half the coefficient of the second term. Ex. 1. Let it be required to solve the equation x2-10x=-16. Completing the square by adding to each member the square of half the coefficient of the second term, we have x2-10x+25=25—16=9. Extracting the root, Whence x—5—±3. x=5±3= 8 or 2, Ans. To verify these values of x, substitute them in the original equation, and we shall have Also, 82-10x8-64-80-16. 22-10×24-20-16. Ex. 2. Solve the equation 2x2+8x—20—70. Ex. 3. Solve the equation 3x2-3x+6=5}. Reducing, Ans. x=+5 or −9. x2 1 Completing the square, x2-x+1 = 1 − 3 = 3% · Second Method of completing the Square. 260. The preceding method of completing the square is always applicable; nevertheless, it sometimes gives rise to inconvenient fractions. In such cases the following method may be preferred. Let the equation be reduced to the form ax2 + bx=c, in which a and b are whole numbers, and prime to each other, but c may be either entire or fractional. Multiply each member of this equation by 4a, and it becomes 4a2x2+4abx=4ac. Adding b2 to each member, we have 4a2x2+4abx+b2=4ac+b2, where the first member is a complete square, and its terms are entire. Extracting the square root, we have 2ax+b=±√4ac+b2. which is the same result as would be obtained by the former rule; but by this method we have avoided the introduction of fractions in completing the square. b 2 Ifb is an even number, will be an entire number; and it would have been sufficient to multiply each member by a, and b2 add to each member. Hence we have the following 4 RULE. 1st. Reduce the equation to the form ax2+bx=c, where a and b are prime to each other. 2d. If b is an odd number, multiply the equation by four times the coefficient of x2, and add to each member the square of the coefficient of x. 3d. If b is an even number, multiply the equation by the coefficient of x2, and add to each member the square of half the coefficient of x. Ex. 4. Solve the equation 6x2-13x=-6. Multiplying by 4×6, and adding 132 to each member, we 144x2-312x+169 169–144–25. have Ex. 5. Solve the equation 110x2-21x=-1. Multiplying by 440, and adding 212 to each member, we have Ex. 6. Solve the equation 7x2-3x=160. 261. Modification of the preceding Method. The preceding method sometimes gives rise to numbers which are unneces sarily large. When the equation has been reduced to the form ax2+bx=c, it is sufficient to multiply it by any number which will render the first term a perfect square. Let the resulting equation be m2x2+nx=q. The first member will become a complete square by the ad Having reduced the equation to the form ax2+bx=c, multiply the equation by any quantity (the least possible) which will render the first term a perfect square. Divide the coefficient of x in this new equation by twice the square root of the coefficient of x2, and add the square of this result to both members. Ex. 7. Solve the equation 8x2+9x=99. Ex. 8. Solve the equation 16x2-15x=34. 16x2-15x+ 15\2 2401 64 17 4x=8 or 4? 17 x=2 or Ans. 16' Ex. 9. Solve the equation 12x2-21+1x. Solve the following equations: Ex. 10. x2-x+201423. Ex. 11. x2-x-40-170. Ex. 12. 3x2+2x-9=76. Ans. x=7 or -61. Ans. x=15 or -14. This equation reduces to x2-15x=-46. Ans. x 11 or 4. Ans. x=3 or 6. Ex. 23. (x-1)(x-2)+(x−2)(x-4)=6(2x-5). Ans. x=8 or 170 170 51 Ex. 24. Ans. x 4 or -13. х x+1x+2 Equations which may be solved like Quadratics. 262. There are many equations of a higher degree than the second, which may be solved by methods similar to those employed for quadratics. To this class belong all equations which contain only two powers of the unknown quantity, and in which |