Elements of Algebra

242 ELEMENTS SECT. IV. $y^2 = \frac{3+3\sqrt{-3}}{2}$, the sum of the squares $x^2 + y^2 = 3$, all the same quantity as required. 688. We may greatly abridge this calculation by a particular artifice, which is applicable likewise to other cases; and which consists in expressing the numbers sought by the sum and the difference of two letters, instead of representing them by distinct letters. In our last question, let us suppose one of the numbers sought to be $p+q$, and the other $p-q$, then their sum will be $2p$, their product will be $p^2-q^2$, and the sum of their squares will be $2p^2+2q^2$, which three quantities must be equal to each other; therefore making the first equal to the second, we have $2p = p^2-q^2$, which gives $q^2 = p^2-2p$. Substituting this value of $q^2$ in the third quantity ($2p^2+2q^2$), and comparing the result $4p^2-4p$ with the first, we have $2p = 4p^2-4p$, whence $p=\frac{3}{2}$. Consequently, $q^2 = p^2 - 2p = -\frac{3}{4}$, and $q = \frac{\sqrt{-3}}{2}$; so that the numbers sought are $p+q = \frac{3+\sqrt{-3}}{2}$, and $p-q = \frac{3-\sqrt{-3}}{2}$, as before. QUESTIONS FOR PRACTICE. 1. What two numbers are those, whose difference is 15, and half of their product equal to the cube of the less? Ans. 3 and 18. 2. To find two numbers whose sum is 100, and product 2059. Ans. 71 and 29. 3. There are three numbers in geometrical progression: the sum of the first and second is 10, and the difference of the second and third is 24. What are they? Ans. 2, 8, and 32. 4. A merchant having laid out a certain sum of money in goods, sells them again for 24l. gaining as much per cent as the goods cost him: required, what they cost him. Ans. 20l. 5. The sum of two numbers is $a$, their product $b$. Required the numbers. Ans. $\frac{a}{2} \pm \sqrt{-b+\frac{a^2}{4}}$, and $\frac{a}{2} \mp \sqrt{-b+\frac{a^2}{4}}$.