Ray's algebra, part second; an analytical treatise, designed for high schools and colleges

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1857 Excerpt: ... 256 gallons; and then filling the vessel with water, draws off the same number of gallons as before, and so on for four draughts, when there were only 81 gallons of pure wine left. How much wine did he draw each time 1 Ans. 64, 48, 36, and 27 gallons. COMPLETE EQUATIONS OF THE SECOND DEGREE. Art. 230. 1. Let it be required to find the value of x in the equation, x1--6a+9=4. It is evident, from Art. 184, that the first member of this equation is a perfect square. By extracting the square root of both members, we find x--3=±2; Whence, a=3±2=3+2=5, or 3--2=1. Verification. (5)J--6(5)+9=4, that is, 25--30+9=4. (1)--6(l)+9=4, that is, 1--6+9=4. Hence, x has two values, +5, and +1, either of which verifies the equation. 2. Let it be required to find the value of x in the equation, x3--6i=27. If the left member of this equation were a perfect square, we might find the value of x by extracting the square root, as in the preceding example. To ascertain what is necessary to render the first member a perfect square, let us compare it with the square of x--a, which is, x1--2ax+a. We have, x'--6x =27. Prom this we see that 2a corresponds to 6; hence, a corresponds to 3, and a3 to 9. Hence, by adding 9, which is the square of half the coefficient of x, to each member, the equation becomes x3--6 a+9=36. Extracting the square root, 3=±6. Whence, x=3±6=--9, or--3, either of which values of x will verify the equation. Aet. 231. We will now proceed to explain the method of completing the square. Since every complete equation of the second degree (Art. 226) may be reduced to the form, ax1--bx=c; if we divide both sides b c by a, we have x'--x= a a For the sake of simplicity, let-=2», and =q. The equa « a Hon then becomes x+2px=q. (1) If-is negati...