Elements of Algebra

222 ELEMENTS SECT. IV. 4. To divide the number 100 into two such parts, that the sum of their square roots may be 14. Ans. 64, and 36. 5. To find three such numbers, that the sum of the first and second multiplied into the third, may be equal to 63; and the sum of the second and third, multiplied into the first, may be equal to 28; also, that the sum of the first and third, multiplied into the second, may be equal to 55. Ans. 2, 5, 9. 6. What two numbers are those, whose sum is to the greater as 11 to 7; the difference of their squares being 132? Ans. 14, and 8. CHAP. VI. Of the Resolution of Mixt Equations of the Second Degree. 638. An equation of the second degree is said to be mixt, or complete, when three terms are found in it; namely, that which contains the square of the unknown quantity, as $ax^2$; that, in which the unknown quantity is found only in the first power, as $bx$; and, lastly, the term which is composed of only known quantities. And since we may unite two or more terms of the same kind into one, and bring all the terms to one side of the sign =, the general form of a mixt equation of the second degree will be $ax^2 \pm bx \pm c = 0$. In this chapter, we shall shew how the value of $x$ may be derived from such equations: and it will be seen, that there are two methods of obtaining it. 639. An equation of the kind that we are now considering may be reduced, by division, to such a form, that the first term will contain only the square, $x^2$, of the unknown quantity $x$. We shall leave the second term on the same side with $x$, and transpose the known term to the other side of the sign =. By these means our equation will assume the form of $x^2 + px = \pm q$, in which $p$ and $q$ represent any known numbers, positive or negative; and the whole is at present reduced to determining the true value of $x$. We shall begin by remarking, that if $x^2 + px$ were a real square, the resolution would be attended with no difficulty, because it would only be required to take the square root of both sides.