CHAPTER VIII. EQUATIONS OF THE SECOND DEGREE. EQUATIONS CONTAINING ONE UNKNOWN QUANTITY. 151. An equation of the second degree containing but one unknown quantity is one in which the greatest exponent of the unknown quantity is equal to 2. Thus, x2 = a, ax2 + bx = c, are equations of the second degree. 152. Let us see to what form every equation of the second degree may be reduced. Take any equation of the second degree, as (1 + x)2 − 2 x − 10 = 5 −2+2 3 4 4 Clearing of fractions, and performing indicated operations, we have 4+8x+4x2 - 3x — 40 = 20 − x + 2x2. Transposing the unknown terms to the first member, the known terms to the second, and arranging with reference to the powers of x, we have 4x2-2x2+8x-3x+x=20+ 40-4, By reducing, 2x2+6x=56. Dividing by the coefficient of x2, we have x+3x28. If we denote the coefficient of x by 2p, and the second member by q, we have x2+2px = q. This is called the reduced equation. 153. When the reduced equation is of this form, it contains three terms, and is called a complete equation. The terms FIRST TERM. The second power of the unknown quantity, with a plus sign. SECOND TERM. with a coefficient. The first power of the unknown quantity, THIRD TERM. A known term in the second member. Every equation of the second degree may be reduced to this form by the following rule: : Clear the equation of fractions, and perform all the indicated operations. Transpose all the unknown terms to the first member, and all the known terms to the second member. Reduce all the terms containing the square of the unknown quantity to a single term, one factor of which is the square of the unknown quantity; reduce also all the terms containing the first power of the unknown quantity to a single term. Divide both members of the resulting equation by the coefficient of the square of the unknown quantity. 154. A root of an equation is such a value of the unknown quantity as, being substituted for it, will satisfy the equation; that is, make the two members equal. The solution of an equation is the operation of finding its roots. INCOMPLETE EQUATIONS. 155. It may happen that 2p, the coefficient of the first power of x, in the equation x+2pxq, is equal to 0. In this case the first power of x will disappear, and the equation will take the form x2 This is called an incomplete equation. Hence (1) An incomplete equation, when reduced, contains but two terms, the square of the unknown quantity, and a known term. 156. Extracting the square root of both members of Equation (1), we have Hence, for the solution of incomplete equations, Reduce the equation to the form x2 = q. Then extract the square root of both members. NOTE. (2) There will be two roots, numerically equal, but having contrary signs. Denoting the first by x', and the second by x', we have 1. What are the values of x in the equation 3x2+8=5x3—10? 2. What are the roots of the equation 3x2+6=4x2 - 10? 3. What are the roots of the equation x2 8 = +10? 9 Ans. x'9, x"=-9. 4. What are the roots of the equation 4x2+13 — 2x2 = 45? Ans. x4, x" — — 4. 5. What are the roots of the equation 6x2 -7=3x2+5? Ans. x'=+2, x": x"=— 2. 6. What are the roots of the equation 8+5x2- +4x2+28? 5 8. What are the roots of the equation x2+ab = 5 x2 ? 9. What are the roots of the equation x√a+x2= b + x2? 1. What number is that which being multiplied by itself the product will be 144? It is plain that the value of x will be found by extracting the square root of both members of the equation; that is, √x2=√144; that is, x = 12. D. N. E. A. 14. 2. A person, being asked how much money he had, said, "If the number of dollars be squared and 6 be added, the sum will be 42." How much had he? Let Ans. $6. x= the number of dollars. 3. A grocer, being asked how much sugar he had sold to a person, answered, "If the square of the number of pounds. be multiplied by 7, the product will be 1575." How many pounds had he sold? Ans. 15. 4. A person, being asked his age, said, "If from the square of my age in years you take 192 years, the remainder will be square of half my age." What was his age? the Ans. 16 years. |