Elements of Algebra

224 ELEMENTS SECT. IV. guishing the following cases: 1. When $p$ is an even number; 2. When $p$ is an odd number; and 3. When $p$ is a fractional number. 1st, Let $p$ be an even number, and the equation such, that $x^2 = 2px+q$; we shall, in this case, have $x = p \pm \sqrt{(p^2+q)}$. 2d, Let $p$ be an odd number, and the equation $x^2 = px+q$; we shall here have $x = \frac{1}{2}p \pm \sqrt{(\frac{1}{4}p^2+q)}$; and since $\frac{1}{4}p^2+q = \frac{p^2+4q}{4}$, we may extract the square root of the denominator, and write $x = \frac{1}{2}p \pm \frac{\sqrt{(p^2+4q)}}{2} = \frac{p \pm \sqrt{(p^2+4q)}}{2}$. 3d, Lastly, if $p$ be a fraction, the equation may be resolved in the following manner. Let the equation be $ax^2 = bx+c$, or $x^2 = \frac{bx}{a} + \frac{c}{a}$, and we shall have, by the rule, $x = \frac{b}{2a} \pm \sqrt{(\frac{b^2}{4a^2} + \frac{c}{a})}$. Now, $\frac{b^2}{4a^2} + \frac{c}{a} = \frac{b^2+4ac}{4a^2}$, the denominator of which is a square; so that $x = \frac{b \pm \sqrt{(b^2+4ac)}}{2a}$. 645. The other method of resolving mixt quadratic equations is, to transform them into pure equations; which is done by substitution: for example, in the equation $x^2 = px+q$, instead of the unknown quantity $x$, we may write another unknown quantity, $y$, such, that $x = y + \frac{1}{2}p$; by which means, when we have determined $y$, we may immediately find the value of $x$. If we make this substitution of $y + \frac{1}{2}p$ instead of $x$, we have $x^2 = y^2 + py + \frac{1}{4}p^2$, and $px = py + \frac{1}{2}p^2$; consequently, our equation will become $y^2 + py + \frac{1}{4}p^2 = py + \frac{1}{2}p^2 + q$; which is first reduced, by subtracting $py$, to $y^2 + \frac{1}{4}p^2 = \frac{1}{2}p^2 + q$; and then, by subtracting $\frac{1}{4}p^2$, to $y^2 = \frac{1}{4}p^2 + q$. This is a pure quadratic equation, which immediately gives $y = \pm \sqrt{(\frac{1}{4}p^2 + q)}$. Now, since $x = y + \frac{1}{2}p$, we have $x = \frac{1}{2}p \pm \sqrt{(\frac{1}{4}p^2 + q)}$,