By Harley Flanders and Justin J. Price (Auth.)

Algebra and Trigonometry

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3. , 3x 2 - 5x = 6x + 1, (JC + 3)(2x - 9) = x2 - Ax, 1*2 + 8 = _ A . 2 + 2x4-i 58 2. EQUATIONS AND INEQUALITIES Simple quadratic equations such as x2 + 1 = 0, x2 = 0, x2 - 1 = 0 show that there may be no real solutions, or one solution, or two solutions. We shall see in this chapter that there are no other possibilities. Solution by Factoring This method depends on a basic property of real numbers: If r and s are real numbers such that rs = 0, then either r = 0, or s = 0, or both. Put in different words, the product of non-zero numbers cannot be zero.

Vx~^T = 1 19. (x2 - 4)(x2 - 9) = 0 18. 1/x2 = 9 20. x2 = x. 15. JC(S - 2)(JC - 7) = 0 16. 4(JC + 1) 2x2 + 1 = 3JC2 - 8 - x 2. LINEAR EQUATIONS In this section and the next, we study two of the simplest and most common types of equations, linear and quadratic equations. A linear equation is one of the form ax + b = 0 (a φ 0), or one that can be brought into that form by moving all terms to one side of the equation. For example, 2x + 1 = - 7 , x - \ = 2x + 1 are linear equations. x+ 8 = 0 , x + f = 0 by shifting terms.

Therefore, it holds just as well if x and y are replaced by any expressions that represent real numbers. Examples: (a - b)2 = a2 - lab + b29 (2a + 3b)2 = 4a2 + \2ab + 9b2, (x + i) 2 = x2 + x + \. In practice, it helps to be able to recognize such squares of sums. For example, given the right sides of the equations above, to realize that they are the squares of the expressions on the left. In hunting for such a perfect square, you need three terms: two squares plus twice the unsquared quantities.