Page:Elementary algebra (1896).djvu/254

236 Rh contains only the square of the unknown it is said to be a pure quadratic.

Thus 2x^2 — 5x = 3 is an affected quadratic, and 5x^2 = 20 is a pure quadratic.

PURE QUADRATIC EQUATIONS.

283. A pure quadratic may be considered as a simple equation in which the square of the unknown quantity is to be found.

Ex. Solve {9}{x^2 -27 } = {25}{x2 - 11 }.

Multiplying across, 9 x^2 — 99 = 25 x^2 — 675 ;

transposing, 16 x^2 = 576; x^2 = 36; and taking the square root of these equals, we have x=\pm6.

Note. We prefix the double sign to the number on the right-hand side for the reason given in Art. 196.

284. In extracting the square root of the two sides of the equation x^2 = 36, it might seem that we ought to prefix the double sign to the quantities on both sides, and write x= ±6. But an examination of the various cases shows this to be unnecessary. For ± x = ± 6 gives the four cases : +x = +6, +x = — 6, —x = +6, — x = — 6, and these are all included in the two already given, namely, x=+6, x = — 6. Hence when we extract the square root of the two sides of an equation, it is sufficient to put the double sign before the square root of one side.

EXAMPLES XXVI. a.

Solve the following equations :

1. 4x^2 + 5=x^2 + 17. 2. 3x^2 + 3= {2x^2}{3} + 24 3. (x+ 1)(x-1) = 2x^2 - 4. 4. {2x^2-6}{2} - {x^2-4}{4} -  {5x^2-10}{7} = 0. 5. x^2 + 2 = {(x -1)^2 - x +24}{x + 2}.