Solving Quadratic Equations

Strategy The setup is identical with that in Example 13.7. However, you will find, on solving for x, that x > 0.050a, so the approximation a — x a fails. The simplest way to proceed is to use the calculated value of x to obtain a better estimate of [HNOJ, then solve again for [H+], An alternative is to use the quadratic formula. (This is a particularly shrewd choice if you have a calculator that can be programmed to solve quadratic equations.)... 

Note There are always two solutions when solving quadratic equations, but only one is meaningful in this type of chemical problem. The other solution, -5.79 x 10-2, is discarded because a negative value for concentration has no physical meaning in this problem. 

Note There are always two solutions when solving quadratic equations, but only one is meaningful in this... 

Ballpark Check Arithmetic errors in solving quadratic equations are common, so it s a good idea to check that the value of x obtained from the quadratic is reasonable. If the approximate value of x (4.2 X 10 3) is fairly small compared to the initial concentration of the acid (0.050 M), as is the case in this problem, then the value of x obtained from the quadratic (4.0 X 10 3) should be fairly close to the approximate value of x. The approximate and more exact values of x agree. 

Solving Quadratic Equations Using the Quadratic Formula... 

The zero product rule is useful when solving quadratic equations because you can rewrite a quadratic equation as equal to zero and take advantage of the fact that one of the factors of the quadratic equation is thus equal to 0. 

In high school algebra you frequently used factoring to solve quadratic equations. However, in this and virtually all other real problems you will encounter in chemistry, neither the numerical coefficients nor the solutions will turn out to be integers, so factoring is not useful. The general solutions to the equation ax2 + bx + c = 0 are given by the expression... 

In the course of solving problems related to equilibria (Frame 47) it is sometimes necessary to solve quadratic equations (i.e. determine their roots or solutions). In order to determine x as the solution of an equation in the quadratic form ... 

A second method for solving quadratic equations is by successive approximations, a systematic method of trial and error. A value of % is guessed and substituted into the equation everywhere % (or x2) appears, except for one place. For example, for the equation... 

Problem-Solving Tip Solving Quadratic Equations—Which Root... 

As you construct and solve quadratic equations based on the observed behavior of matter, you must decide which root has physical significance. Examination of the equation that defines x always gives clues about possible values for x. In this way you can tell which is extraneous (has no physical significance). Negative roots are often extraneous. 

The first solution is physically impossible since the amounts of Ha and la reacted would be more than those originally present. The second solution gives the correct answer. Note that in solving quadratic equations of this type, one answer is always physically impossible, so choosing a value for x is easy. Step 3 At equilibrium, the concentrations are... 

Take note Quadratic equations always have two roots. When solving quadratic equations a check should always be made to see whether both solutions are physically possible. 

Take note when solving quadratic equations a check must be made to see whether both solutions are physically possible. Often it will be found that one solution will give either a concentration which is negative or, as here, one which is greater than the initial concentration. The analogue of Equation (7.13) for Mg " (aq) is ... 

Quadratic equations take the standard form ax2 + bx + c = 0. The most appropriate method of solving quadratic equations in scientific problems is the use of the quadratic formula. The quadratic formula produces the solutions of a standard form quadratic equation. 

A concentration can only be positive, and so the negative value of x is not a solution You can use Excel Solver for solving quadratic equations. See Chapter 6. 

In addition to basic arithmetic and algebra, four mathematieal operations are used frequently in general chemistry manipulating logarithms, using exponential notation, solving quadratic equations, and graphing data. Eaeh is diseussed briefly below. 

Kaplan (1996) has also provided an excellent example of knowledge abstraction. Quadratic equations can always be solved using the method of trial and error. However, each algebra book provides two formulas or abstract solutions that can be used to solve any quadratic equation. We could say that these abstract solutions represent our abstract and universal knowledge about solving quadratic equations that is, they are applicable to any quadratic equations. 

Using the fomiula for solving quadratic equations allows deriving the following... 

One can prove that every fraction has either a finite or an Infinite, but periodic number of decimal places. All rational numbers fill the number line densely with an Infinity of points. Between these densely arranged points, one finds the also Infinite number of "Irrational numbers" which are represented by an infinite, not periodic number of decimal places and can thus not be represented by fractions (such as e = 2.7182..., TT = 3.1415..., orT " 1.4142...). Since the rational numbers are densely placed, one can always approximate the irrational numbers with unlimited precision by rational numbers. The total of rational and Irrational numbers are called the real numbers. The real numbers must be expanded with the complex numbers which occur, for example, in efforts to solve quadratic equations. [I.e., (x-5) -4 has the solution x = 5 2i], For complex numbers see also Sect. 2.3.5 and Fig. A.6.3. 

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