![]() This makes sense chemically, hut one is left with an uneasy feeling that the quadratic equation ought to he able to yield a satisfactory solution. I t is therefore usual to adopt the approximation that s is much smaller than 0.1, in order to avoid solving the quadratic. The standard solution does not work well, if a t all, with a hand calculator. The equilibrium constant expression, hearing in mind the common ion, is: S(S + 0.1) = 8 X + - which can he rearranged to s2 0.1s 8 X 10-l7 = 0. As anexample, consider the calculation of the soluhility,~, of silver iodide ( K, = 8 X 10-17) in 0.1 F sodium iodide. If one of the roots is very small, its value is determined through eq 4 hy division of one numher by another much larger numher, rather than by suhtraction i f two nearly equal quantities as a result, there is no loss of significant figures. Then the two roots of the quadratic equation (eq 1) can be written: In calculating q, the square root appears with the same sign as b itself, and so the damaging suhtraction used in obtaining one of the roots from eq 2 is replaced (in hoth roots) by an addition of quantities with the same sign. A much better technique that avoids round-off error of this sort has been described.' Define the quantity q as follows: where sgn(b) means "the sign of b". This situation can occur in calculations related to chemical equilibrium. A recent hook on the ~racticalaspects of numerical analysis' points out that this formula can iead to unacceptable roundoff error when 4ac is much less than b2, for then the square root is very close to b and one of the roots is obtained by suhtraction of b. ![]() The roots of the equation a~~+bx+c=O (1) = -O.lOWW, to sufficient accuracy The solution to the problem is: are almost always calculated from the formula: This formula is often used without explanation in chemistry classes and has been memorized by &erations of student. Canada Solving quadratic equations is a piece of algebra that is usually taken for granted in chemistry classes. Brown Queen's University, Kingston, ON K7L 3N6. ![]()
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