109. Subjoined are a few examples for practice in the transformation and reduction of expressions involving √ −1. (1) √—48=√(−16 × 3)=4.√3.√—1. (2) (a+b√−1)±(c+d√−1)=a±c+(b±d)√ — 1. (3) (a√−c)(b√—d)=ab√(cd) (√—1)2=—ab √(cd). (4) {a+√(-4b2)} {a2—3√(−b1)}={a+2b√(−1)} {a2—3b2√(−1)} =a3+(2a2b-3ab2) √−1+6b3. (5) {x+√(-4y)} {x—√(—9y)}=x2-- x √ √―1+6y. (6) (a+b√−1)3— (a—b √/—1)3—(6a2b—b3) √ — 1. (7) (a + b√ 1 + c √ − 1) (a + b √ − 1 − c √ − 1) =(a+b√−1)2+c2=a2—b2+c2+2ab √—1. (10) (5+2 √−1) (6—3 √✓ −1)=36—3 √−1. EQUATIONS. 110. The term Equation is generally applied to any expressions, including zero, which are connected by the sign. 111. In the theory of equations those only are considered which contain one or more unknown quantities; the object being to determine, from the relations expressed between the known and unknown quantities, the values of the latter in terms of the former. These values are termed the roots of the equation, and are said to satisfy the equation, because, when substituted for the unknown quantities, they render its members identical. 112. The general principle by means of which a root of an equation may be discovered is, that any change may be made in its members which does not affect their equality. Hence, (1) The same quantity may be added to, or subtracted from, both sides of an equation. (2) Any term may be transferred from one side of an equation to the other, its sign being changed: for if a positive term be thus transferred, it is equivalent to subtracting the same quantity from both sides of the equation; and if a negative quantity be transferred, it is equivalent to adding the same quantity to both sides. (3) Both members of an equation may be multiplied or divided by the same or equal quantities. Hence, an equation may be cleared of fractions by multiplying both its members by each of the denominators, or by the least common multiple of all the denominators. Also, the signs of all the terms on both sides may be changed, since this will only be introducing into each term the factor-1. Hence also, both members of an equation may be raised to any power; and conversely, any root of both may be extracted. 113. When an equation is cleared of fractions and surds, if it contain the first power only of an unknown quantity, it is called a simple equation, or an equation of the first degree: if it contain the square of an unknown quantity, it is called a quadratic, or an equation of the second degree; and generally, if the index of the highest power of an unknown quantity in any equation be n, whether the inferior powers be involved or not, it is called an equation of the nth degree. SIMPLE EQUATIONS INVOLVING ONE 114. Every simple equation containing but one unknown quantity may be put under the form x-p0, or x = p: whence it is obvious that it can have but one root. |