PROPERTIES OF QUADRATIC SURDS. 250. A Quadratic Surd is the indicated square root of an imperfect square; as, √3, or √7. 251. A quadratic surd cannot be equal to a rational quantity plus a quadratic surd. Squaring the equation, a = b2 + 2 bc + c. That is, a surd equal to a rational quantity, which is impossible.. Hence va cannot be equal to b + √c. 252. To prove that if a + b = c + √d, then a = c, and b = √d. If a is not equal to c, let a = c +x. Substituting, we have which is impossible by Art. 251. Hence a = c, and consequently b = √d. 253. To prove that if √a + √b =√x+√y, then √a - √b =√x - √y. Extracting the square root, Va - √b = √x - √y. (1) (2) SQUARE ROOT OF A BINOMIAL SURD. 254. The preceding principles serve to extract the square root of a binomial surd whose first term is rational. For example, required the square root of 13 - √160. √13-√160 = √x-Vy. (1) Then, by Art. 253, √13+√160=√x+√y. (2) Multiplying (1) by (2), √169-160 = х - у. 255. Examples like the above may often be solved by inspection by expressing the given quantity in the form of a perfect trinomial square (Art. 108), as follows: Reduce the surd term so that its coefficient may be 2. Separate the rational term into two parts whose product is the quantity under the radical sign. Extract the square roots of these parts, and connect them by the sign of the surd term. 1. Extract the square root of 8 + √48. √8+√48=√8+ 2√12. We then separate 8 into two parts whose product is 12. The parts are 6 and 2; hence, √8+2√12=6+2√6×2+2 2. Extract the square root of 22 - 3√32. 22 3 √32 = √22 - √288 =√22 - 2√72. We then separate 22 into two parts whose product is 72. The parts are 18 and 4; hence, 2. Solve the equation √2x-1+√2x+6=7. Transpose the terms of the equation so that a radical term may stand alone in one member; then raise both members to a power of the same degree as the radical. If there are still radical terms remaining, repeat the operation. Note. The equation should be simplified as much as possible before performing the involution. |