Elements of Algebra

CHAP. X. OF ALGEBRA. 251 when $x^2 + cx + c^2 = 0$. Now, this expression contains two other values of $x$; for it gives $x^2 = -cx - c^2$, and $x = -\frac{c}{2} \pm \sqrt{(\frac{c^2}{4} - c^2)}$, or - - - - - - - $x = \frac{-c \pm \sqrt{-3c^2}}{2}$; that is to say, $x = \frac{-c \pm c\sqrt{-3}}{2} = \frac{-1 \pm \sqrt{-3}}{2} \times c$. 713. Now, as $c$ was substituted for $\sqrt[3]{a}$, we conclude, that every equation of the third degree, of the form $x^3 = a$, furnishes three values of $x$ expressed in the following manner: 1. $x = \sqrt[3]{a}$, 2. $x = \frac{-1+\sqrt{-3}}{2} \times \sqrt[3]{a}$, 3. $x = \frac{-1-\sqrt{-3}}{2} \times \sqrt[3]{a}$. This shews, that every cube root has three different values; but that one only is real, or possible, the two others being impossible. This is the more remarkable, since every square root has two values, and since we shall afterwards see, that a biquadratic root has four different values, that a fifth root has five values, and so on. In ordinary calculations, indeed, we employ only the first of those values, because the other two are imaginary; as we shall shew by some examples. 714. Question 1. To find a number, whose square, multiplied by its fourth part, may produce 432. Let $x$ be that number; the product of $x^2$ multiplied by $\frac{1}{4}x$ must be equal to the number 432, that is to say, $\frac{1}{4}x^3 = 432$, and $x^3 = 1728$; whence, by extracting the cube root, we have $x = 12$. The number sought therefore is 12; for its square 144, multiplied by its fourth part, or by 3, gives 432. 715. Question 2. Required a number such, that if we divide its fourth power by its half, and add $14\frac{1}{4}$ to the product, the sum may be 100. Calling that number $x$, its fourth power will be $x^4$; dividing by the half, or $\frac{1}{2}x$, we have $2x^3$; and adding to that $14\frac{1}{4}$, the sum must be 100. We have therefore $2x^3 + 14\frac{1}{4} = 100$; subtracting $14\frac{1}{4}$, there remains $2x^3 = \frac{343}{4}$; dividing by 2, gives $x^3 = \frac{343}{8}$; and extracting the cube root, we find $x = \frac{7}{2}$. 716. Question 3. Some officers being quartered in a