12. The sum of two numbers is a, and the sum of their reciprocals is b. Required the numbers. General Properties of Equations of the Second Degree. 272. Every equation of the second degree containing but one unknown quantity has two roots, and only two. We have seen, Art. 250, that every equation of the second degree containing but one unknown quantity can be reduced to the form x2+px=q. We have also found, Art. 257, that this equation has two roots, viz., This equation can not have more than two roots; for, if possible, suppose it to have three roots, and represent these roots by x', x', and x"". Then, since a root of an equation is such a number as, substituted for the unknown quantity, will satisfy the equation, we must have that is, the third supposed root is identical with the second; hence there can not be three different roots to a quadratic equation. 273. The algebraic sum of the two roots is equal to the coefficient of the second term of the equation taken with the contrary sign. X If we add together the two values of x in the general equation, the radical parts having opposite signs disappear, and we obtain p. Thus, in the equation x2-10x=-16, the two roots are 8 and 2, whose sum is +10, the coefficient of x taken with the contrary sign. If the two roots are equal numerically, but have opposite signs, their sum is zero, and the second term of the equation vanishes. Thus the two roots of the equation x2=16 are +4 and -4, whose sum is zero. This equation may be written x2+0x=16. 274. The product of the two roots is equal to the second member of the equation taken with the contrary sign. If we multiply together the two values of x (observing that the product of the sum and difference of two quantities is equal to the difference of their squares), we obtain Thus, in the equation x2-10x=-16, the product of the two roots 8 and 2 is +16, which is equal to the second member of the equation taken with the contrary sign. 275. The last two principles enable us to form an equation whose roots shall be any given quantities. Ex. 1. Find the equation whose roots are 3 and 5. According to Art. 273, the coefficient of the second term of the equation must be -8; and, according to Art. 274, the second member of the equation must be -15. Hence the equation is Ex. 2. Find the equation whose roots are x2-8x=-15. 4 and 7. Ex. 3. Find the equation whose roots are 5 and -9. Ex. 4. Find the equation whose roots are 6 and +11. Ex. 7. Find the equation whose roots are — and +1. and +1. Ex. 8. Find the equation whose roots are 1±√5. 276. Every equation of the second degree whose roots are a and b, may be reduced to the form (x-a) (x—b)=0. Thus the equation x2-10x=-16, whose roots are 8 and 2, may be resolved into the factors x-8=0 and x-2=0. It is also obvious that if a is a root of an equation of the second degree, the equation must be divisible by x-a. Thus the preceding equation is divisible by x-8, giving the quotient x-2. Ex. 1. The roots of the equation x2+6x+8=0 are -2 and -4. Resolve the first member into its factors. Ex. 2. The roots of the equation x2+6x-27=0 are +3 and -9. Resolve the first member into its factors. Ex. 3. The roots of the equation x2-2x-24-0 are +6 and -4. Resolve the first member into its factors. Ex. 4. Resolve the equation a2+73x+780-0 into simple factors. Ans. (x+60) (x+13)=0. Ex. 5. Resolve the equation x2-88x+1612-0 into simple factors. Ans. (x-62) (x-26)=0. Ex. 6. Resolve the equation 2x2+x-6=0 into simple facAns. 2(x+2)(x—3)=0. tors. Ex. 7. Resolve the equation 3x2-10x-25-0 into simple Ans. 3(x-5) (x+3)=0. factors. Discussion of the General Equation of the Second Degree. 277. In the general equation of the second degree x2+px=q, the coefficient of x, as well as the absolute term, may be either positive or negative. We may therefore have the four following forms: We will now consider what conditions will render these roots positive or negative, equal or unequal, real or imaginary. 厚 p2 For the same reason, -9 must be less than 2. 4 2* Therefore, in the first and second forms, the sign of the roots will correspond to the sign of the radicals; but in the third and fourth forms the sign of the roots will correspond to the sign of the rational parts. Hence, in the first form, one root is positive and the other negative, and the negative root is numerically the greatest; as in the equation x2+x=6, whose roots are +2 and -3. In the second form one root is positive and the other negative, and the positive root is numerically the greatest, as in the equation x2 x=6, whose roots are -2 and +3. In the third form both roots are negative, as in the equation x2+5x=-6, whose roots are -2 and -3. +5х. In the fourth form both roots are positive, as in the equation x2-5x=-6, whose roots are +2 and +3. 279. Equal and unequal roots. In the first and second forms the two roots are always unequal; but in the third and fourth forms, when q is numerically equal p2 to the radical part of both values of x becomes zero, and 4' the two roots are then said to be equal. In this case Thus, in the equation x2+6x=-9, the two roots are −3 and -3. We say that in this case the equation has two roots, because it is the product of the two factors x+3=0 and x+3=0. So, also, in the equation x2-6x=-9, the two roots are +3 and +3. 280. Real and imaginary roots. Since 22, being a square, is positive for all real values of p, 4' it follows that the expression p2 2 +q can only be rendered neg 4 ative by the sign of q; that is, the quantity under the radical sign can only be negative when q is negative and numerically greater than 2. Hence, in the first and second forms, both roots 202 are always real; but in the third and fourth forms both roots are imaginary when q is numerically greater than 202 |