Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Appendix A.5 Solving Equations A49 A.5 SOLVING EQUATIONS What you should learn • Identify different types of equations. • Solve linear equations in one variable and equations that lead to linear equations. • Solve quadratic equations by factoring, extracting square roots, completing the square, and using the Quadratic Formula. • Solve polynomial equations of degree three or greater. • Solve equations involving radicals. • Solve equations with absolute values. • Use common formulas to solve reallife problems. Why you should learn it Linear equations are used in many real-life applications. For example, in Exercises 155 and 156 on pages A61 and A62, linear equations can be used to model the relationship between the length of a thigh bone and the height of a person, helping researchers learn about ancient cultures. Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. For example 3x ⫺ 5 ⫽ 7, x 2 ⫺ x ⫺ 6 ⫽ 0, and 冪2x ⫽ 4 are equations. To solve an equation in x means to find all values of x for which the equation is true. Such values are solutions. For instance, x ⫽ 4 is a solution of the equation 3x ⫺ 5 ⫽ 7 because 3共4兲 ⫺ 5 ⫽ 7 is a true statement. The solutions of an equation depend on the kinds of numbers being considered. For instance, in the set of rational numbers, x 2 ⫽ 10 has no solution because there is no rational number whose square is 10. However, in the set of real numbers, the equation has the two solutions x ⫽ 冪10 and x ⫽ ⫺ 冪10. An equation that is true for every real number in the domain of the variable is called an identity. The domain is the set of all real numbers for which the equation is defined. For example x2 ⫺ 9 ⫽ 共x ⫹ 3兲共x ⫺ 3兲 Identity is an identity because it is a true statement for any real value of x. The equation 1 x ⫽ 3x2 3x Identity where x ⫽ 0, is an identity because it is true for any nonzero real value of x. An equation that is true for just some (or even none) of the real numbers in the domain of the variable is called a conditional equation. For example, the equation x2 ⫺ 9 ⫽ 0 Conditional equation is conditional because x ⫽ 3 and x ⫽ ⫺3 are the only values in the domain that satisfy the equation. The equation 2x ⫺ 4 ⫽ 2x ⫹ 1 is conditional because there are no real values of x for which the equation is true. Linear Equations in One Variable Definition of a Linear Equation A linear equation in one variable x is an equation that can be written in the standard form ax ⫹ b ⫽ 0 where a and b are real numbers with a ⫽ 0. A50 Appendix A Review of Fundamental Concepts of Algebra A linear equation in one variable, written in standard form, always has exactly one solution. To see this, consider the following steps. Note that some linear equations in nonstandard form have no solution or infinitely many solutions. For instance, x⫽x⫹1 has no solution because it is not true for any value of x. Because 5x ⫹ 10 ⫽ 5共x ⫹ 2兲 is true for any value of x, the equation has infinitely many solutions. ax ⫹ b ⫽ 0 Original equation, with a ⫽ 0 ax ⫽ ⫺b x⫽⫺ b a Subtract b from each side. Divide each side by a. To solve a conditional equation in x, isolate x on one side of the equation by a sequence of equivalent (and usually simpler) equations, each having the same solution(s) as the original equation. The operations that yield equivalent equations come from the Substitution Principle (see Appendix A.1) and simplification techniques. Generating Equivalent Equations An equation can be transformed into an equivalent equation by one or more of the following steps. Given Equation 2x ⫺ x ⫽ 4 Equivalent Equation x⫽4 2. Add (or subtract) the same quantity to (from) each side of the equation. x⫹1⫽6 x⫽5 3. Multiply (or divide) each side of the equation by the same nonzero quantity. 2x ⫽ 6 x⫽3 4. Interchange the two sides of the equation. 2⫽x x⫽2 1. Remove symbols of grouping, combine like terms, or simplify fractions on one or both sides of the equation. Example 1 After solving an equation, you should check each solution in the original equation. For instance, you can check the solution of Example 1(a) as follows. 3x ⫺ 6 ⫽ 0 ? 3共2兲 ⫺ 6 ⫽ 0 0⫽0 Write original equation. Try checking the solution of Example 1(b). a. 3x ⫺ 6 ⫽ 0 Original equation 3x ⫽ 6 Add 6 to each side. x⫽2 Divide each side by 3. b. 5x ⫹ 4 ⫽ 3x ⫺ 8 Substitute 2 for x. Solution checks. Solving a Linear Equation ✓ 2x ⫹ 4 ⫽ ⫺8 2x ⫽ ⫺12 x ⫽ ⫺6 Original equation Subtract 3x from each side. Subtract 4 from each side. Divide each side by 2. Now try Exercise 15. Appendix A.5 An equation with a single fraction on each side can be cleared of denominators by cross multiplying. To do this, multiply the left numerator by the right denominator and the right numerator by the left denominator as follows. a c ⫽ b d ad ⫽ cb A51 To solve an equation involving fractional expressions, find the least common denominator (LCD) of all terms and multiply every term by the LCD. This process will clear the original equation of fractions and produce a simpler equation. Example 2 Solve An Equation Involving Fractional Expressions x 3x ⫹ ⫽ 2. 3 4 Solution 3x x ⫹ ⫽2 3 4 Original equation Cross multiply. Solving Equations Write original equation. x 3x 共12兲 ⫹ 共12兲 ⫽ 共12兲2 3 4 Multiply each term by the LCD of 12. 4x ⫹ 9x ⫽ 24 Divide out and multiply. 13x ⫽ 24 x⫽ Combine like terms. 24 13 Divide each side by 13. The solution is x ⫽ 24 13 . Check this in the original equation. Now try Exercise 23. When multiplying or dividing an equation by a variable quantity, it is possible to introduce an extraneous solution. An extraneous solution is one that does not satisfy the original equation. Therefore, it is essential that you check your solutions. Example 3 Solve An Equation with an Extraneous Solution 3 6x . 1 ⫽ ⫺ x ⫺ 2 x ⫹ 2 x2 ⫺ 4 Solution The LCD is x 2 ⫺ 4, or 共x ⫹ 2兲共x ⫺ 2兲. Multiply each term by this LCD. Recall that the least common denominator of two or more fractions consists of the product of all prime factors in the denominators, with each factor given the highest power of its occurrence in any denominator. For instance, in Example 3, by factoring each denominator you can determine that the LCD is 共x ⫹ 2兲共x ⫺ 2兲. 1 3 6x 共x ⫹ 2兲共x ⫺ 2兲 ⫽ 共x ⫹ 2兲共x ⫺ 2兲 ⫺ 2 共x ⫹ 2兲共x ⫺ 2兲 x⫺2 x⫹2 x ⫺4 x ⫹ 2 ⫽ 3共x ⫺ 2兲 ⫺ 6x, x ⫽ ±2 x ⫹ 2 ⫽ 3x ⫺ 6 ⫺ 6x x ⫹ 2 ⫽ ⫺3x ⫺ 6 4x ⫽ ⫺8 x ⫽ ⫺2 Extraneous solution In the original equation, x ⫽ ⫺2 yields a denominator of zero. So, x ⫽ ⫺2 is an extraneous solution, and the original equation has no solution. Now try Exercise 35. A52 Appendix A Review of Fundamental Concepts of Algebra Quadratic Equations A quadratic equation in x is an equation that can be written in the general form ax2 ⫹ bx ⫹ c ⫽ 0 where a, b, and c are real numbers, with a ⫽ 0. A quadratic equation in x is also known as a second-degree polynomial equation in x. You should be familiar with the following four methods of solving quadratic equations. Solving a Quadratic Equation Factoring: If ab ⫽ 0, then a ⫽ 0 or b ⫽ 0. x2 ⫺ x ⫺ 6 ⫽ 0 Example: 共x ⫺ 3兲共x ⫹ 2兲 ⫽ 0 x⫺3⫽0 x⫽3 x⫹2⫽0 x ⫽ ⫺2 Square Root Principle: If u 2 ⫽ c, where c > 0, then u ⫽ ± 冪c. 共x ⫹ 3兲2 ⫽ 16 Example: The Square Root Principle is also referred to as extracting square roots. x ⫹ 3 ⫽ ±4 x ⫽ ⫺3 ± 4 x ⫽ 1 or x ⫽ ⫺7 Completing the Square: If x 2 ⫹ bx ⫽ c, then 冢2冣 2 冢x ⫹ 2 冣 2 x 2 ⫹ bx ⫹ b b Example: ⫽c⫹ 冢2冣 ⫽c⫹ b2 . 4 b 2 冢b2冣 2 Add 冢62冣 2 Add to each side. x 2 ⫹ 6x ⫽ 5 x 2 ⫹ 6x ⫹ 32 ⫽ 5 ⫹ 32 to each side. 共x ⫹ 3兲 ⫽ 14 2 x ⫹ 3 ⫽ ± 冪14 x ⫽ ⫺3 ± 冪14 Quadratic Formula: If ax 2 ⫹ bx ⫹ c ⫽ 0, then x ⫽ You can solve every quadratic equation by completing the square or using the Quadratic Formula. Example: ⫺b ± 冪b2 ⫺ 4ac . 2a 2x 2 ⫹ 3x ⫺ 1 ⫽ 0 x⫽ ⫽ ⫺3 ± 冪32 ⫺ 4共2兲共⫺1兲 2共2兲 ⫺3 ± 冪17 4 Appendix A.5 Example 4 a. A53 Solving a Quadratic Equation by Factoring 2x 2 ⫹ 9x ⫹ 7 ⫽ 3 2x2 Solving Equations Original equation ⫹ 9x ⫹ 4 ⫽ 0 Write in general form. 共2x ⫹ 1兲共x ⫹ 4兲 ⫽ 0 Factor. 1 2 2x ⫹ 1 ⫽ 0 x⫽⫺ x⫹4⫽0 x ⫽ ⫺4 Set 1st factor equal to 0. Set 2nd factor equal to 0. The solutions are x ⫽ ⫺ 12 and x ⫽ ⫺4. Check these in the original equation. b. 6x 2 ⫺ 3x ⫽ 0 Original equation 3x共2x ⫺ 1兲 ⫽ 0 3x ⫽ 0 2x ⫺ 1 ⫽ 0 Factor. x⫽0 x⫽ 1 2 Set 1st factor equal to 0. Set 2nd factor equal to 0. The solutions are x ⫽ 0 and x ⫽ 12. Check these in the original equation. Now try Exercise 49. Note that the method of solution in Example 4 is based on the Zero-Factor Property from Appendix A.1. Be sure you see that this property works only for equations written in general form (in which the right side of the equation is zero). So, all terms must be collected on one side before factoring. For instance, in the equation 共x ⫺ 5兲共x ⫹ 2兲 ⫽ 8, it is incorrect to set each factor equal to 8. Try to solve this equation correctly. Example 5 Extracting Square Roots Solve each equation by extracting square roots. b. 共x ⫺ 3兲2 ⫽ 7 a. 4x 2 ⫽ 12 Solution a. 4x 2 ⫽ 12 x2 Write original equation. ⫽3 Divide each side by 4. x ⫽ ± 冪3 Extract square roots. When you take the square root of a variable expression, you must account for both positive and negative solutions. So, the solutions are x ⫽ 冪3 and x ⫽ ⫺ 冪3. Check these in the original equation. b. 共x ⫺ 3兲2 ⫽ 7 x ⫺ 3 ⫽ ± 冪7 x ⫽ 3 ± 冪7 Write original equation. Extract square roots. Add 3 to each side. The solutions are x ⫽ 3 ± 冪7. Check these in the original equation. Now try Exercise 65. A54 Appendix A Review of Fundamental Concepts of Algebra When solving quadratic equations by completing the square, you must add 共b兾2兲 2 to each side in order to maintain equality. If the leading coefficient is not 1, you must divide each side of the equation by the leading coefficient before completing the square, as shown in Example 7. Example 6 Completing the Square: Leading Coefficient Is 1 Solve x 2 ⫹ 2x ⫺ 6 ⫽ 0 by completing the square. Solution x 2 ⫹ 2x ⫺ 6 ⫽ 0 Write original equation. x 2 ⫹ 2x ⫽ 6 x2 ⫹ 2x ⫹ 12 Add 6 to each side. ⫽6⫹ 12 Add 12 to each side. 2 共half of 2兲 共x ⫹ 1兲2 ⫽ 7 Simplify. x ⫹ 1 ⫽ ± 冪7 x ⫽ ⫺1 ± 冪7 Take square root of each side. Subtract 1 from each side. The solutions are x ⫽ ⫺1 ± 冪7. Check these in the original equation. Now try Exercise 73. Example 7 Completing the Square: Leading Coefficient Is Not 1 3x2 ⫺ 4x ⫺ 5 ⫽ 0 Original equation 3x ⫺ 4x ⫽ 5 2 Add 5 to each side. 4 5 x2 ⫺ x ⫽ 3 3 冢 冣 4 2 x2 ⫺ x ⫹ ⫺ 3 3 2 ⫽ Divide each side by 3. 冢 冣 5 2 ⫹ ⫺ 3 3 2 Add 共⫺ 3 兲 to each side. 2 2 共half of ⫺ 43 兲2 4 4 19 x2 ⫺ x ⫹ ⫽ 3 9 9 冢x ⫺ 32冣 x⫺ 2 ⫽ 19 9 冪19 2 ⫽ ± 3 3 x⫽ 冪19 2 ± 3 3 Now try Exercise 77. Simplify. Perfect square trinomial Extract square roots. Solutions Appendix A.5 WARNING / CAUTION When using the Quadratic Formula, remember that before the formula can be applied, you must first write the quadratic equation in general form. Example 8 Solving Equations A55 The Quadratic Formula: Two Distinct Solutions Use the Quadratic Formula to solve x 2 ⫹ 3x ⫽ 9. Solution x2 ⫹ 3x ⫽ 9 Write original equation. x ⫹ 3x ⫺ 9 ⫽ 0 2 Write in general form. x⫽ ⫺b ± 冪b2 ⫺ 4ac 2a Quadratic Formula x⫽ ⫺3 ± 冪共3兲2 ⫺ 4共1兲共⫺9兲 2共1兲 Substitute a ⫽ 1, b ⫽ 3, and c ⫽ ⫺9. x⫽ ⫺3 ± 冪45 2 Simplify. x⫽ ⫺3 ± 3冪5 2 Simplify. The equation has two solutions: x⫽ ⫺3 ⫹ 3冪5 2 and x⫽ ⫺3 ⫺ 3冪5 . 2 Check these in the original equation. Now try Exercise 87. Example 9 The Quadratic Formula: One Solution Use the Quadratic Formula to solve 8x 2 ⫺ 24x ⫹ 18 ⫽ 0. Solution 8x2 ⫺ 24x ⫹ 18 ⫽ 0 Write original equation. Divide out common factor of 2. 4x ⫺ 12x ⫹ 9 ⫽ 0 2 x⫽ ⫺b ± 冪b2 ⫺ 4ac 2a Quadratic Formula x⫽ ⫺ 共⫺12兲 ± 冪共⫺12兲2 ⫺ 4共4兲共9兲 2共4兲 Substitute a ⫽ 4, b ⫽ ⫺12, and c ⫽ 9. x⫽ 12 ± 冪0 3 ⫽ 8 2 Simplify. 3 This quadratic equation has only one solution: x ⫽ 2. Check this in the original equation. Now try Exercise 91. Note that Example 9 could have been solved without first dividing out a common factor of 2. Substituting a ⫽ 8, b ⫽ ⫺24, and c ⫽ 18 into the Quadratic Formula produces the same result. A56 Appendix A Review of Fundamental Concepts of Algebra Polynomial Equations of Higher Degree WARNING / CAUTION A common mistake that is made in solving equations such as the equation in Example 10 is to divide each side of the equation by the variable factor x 2. This loses the solution x ⫽ 0. When solving an equation, always write the equation in general form, then factor the equation and set each factor equal to zero. Do not divide each side of an equation by a variable factor in an attempt to simplify the equation. The methods used to solve quadratic equations can sometimes be extended to solve polynomial equations of higher degree. Example 10 Solving a Polynomial Equation by Factoring Solve 3x 4 ⫽ 48x 2. Solution First write the polynomial equation in general form with zero on one side, factor the other side, and then set each factor equal to zero and solve. 3x 4 ⫽ 48x 2 Write original equation. 3x 4 ⫺ 48x 2 ⫽ 0 Write in general form. 3x 2共x 2 ⫺ 16兲 ⫽ 0 Factor out common factor. 3x 2共x ⫹ 4兲共x ⫺ 4兲 ⫽ 0 Write in factored form. x⫽0 Set 1st factor equal to 0. x⫹4⫽0 x ⫽ ⫺4 Set 2nd factor equal to 0. x⫺4⫽0 x⫽4 Set 3rd factor equal to 0. 3x 2 ⫽ 0 You can check these solutions by substituting in the original equation, as follows. Check 3共0兲4 ⫽ 48共0兲 2 3共⫺4兲 ⫽ 48共⫺4兲 4 0 checks. 2 ✓ ⫺4 checks. 3共4兲4 ⫽ 48共4兲 2 4 checks. ✓ ✓ So, you can conclude that the solutions are x ⫽ 0, x ⫽ ⫺4, and x ⫽ 4. Now try Exercise 113. Example 11 Solving a Polynomial Equation by Factoring Solve x 3 ⫺ 3x 2 ⫺ 3x ⫹ 9 ⫽ 0. Solution x3 ⫺ 3x 2 ⫺ 3x ⫹ 9 ⫽ 0 Write original equation. x2共x ⫺ 3兲 ⫺ 3共x ⫺ 3兲 ⫽ 0 Factor by grouping. 共x ⫺ 3兲共x 2 ⫺ 3兲 ⫽ 0 x⫺3⫽0 x2 ⫺ 3 ⫽ 0 Distributive Property x⫽3 Set 1st factor equal to 0. x ⫽ ± 冪3 Set 2nd factor equal to 0. The solutions are x ⫽ 3, x ⫽ 冪3, and x ⫽ ⫺ 冪3. Check these in the original equation. Now try Exercise 119. Appendix A.5 Solving Equations A57 Equations Involving Radicals Operations such as squaring each side of an equation, raising each side of an equation to a rational power, and multiplying each side of an equation by a variable quantity all can introduce extraneous solutions. So, when you use any of these operations, checking your solutions is crucial. Example 12 Solving Equations Involving Radicals a. 冪2x ⫹ 7 ⫺ x ⫽ 2 Original equation 冪2x ⫹ 7 ⫽ x ⫹ 2 Isolate radical. 2x ⫹ 7 ⫽ x 2 ⫹ 4x ⫹ 4 Square each side. 0 ⫽ x 2 ⫹ 2x ⫺ 3 Write in general form. 0 ⫽ 共x ⫹ 3兲共x ⫺ 1兲 Factor. x⫹3⫽0 x ⫽ ⫺3 Set 1st factor equal to 0. x⫺1⫽0 x⫽1 Set 2nd factor equal to 0. By checking these values, you can determine that the only solution is x ⫽ 1. b. 冪2x ⫺ 5 ⫺ 冪x ⫺ 3 ⫽ 1 Original equation 冪2x ⫺ 5 ⫽ 冪x ⫺ 3 ⫹ 1 Isolate 冪2x ⫺ 5. 2x ⫺ 5 ⫽ x ⫺ 3 ⫹ 2冪x ⫺ 3 ⫹ 1 Square each side. 2x ⫺ 5 ⫽ x ⫺ 2 ⫹ 2冪x ⫺ 3 Combine like terms. x ⫺ 3 ⫽ 2冪x ⫺ 3 When an equation contains two radicals, it may not be possible to isolate both. In such cases, you may have to raise each side of the equation to a power at two different stages in the solution, as shown in Example 12(b). Isolate 2冪x ⫺ 3. x 2 ⫺ 6x ⫹ 9 ⫽ 4共x ⫺ 3兲 Square each side. x 2 ⫺ 10x ⫹ 21 ⫽ 0 Write in general form. 共x ⫺ 3兲共x ⫺ 7兲 ⫽ 0 Factor. x⫺3⫽0 x⫽3 Set 1st factor equal to 0. x⫺7⫽0 x⫽7 Set 2nd factor equal to 0. The solutions are x ⫽ 3 and x ⫽ 7. Check these in the original equation. Now try Exercise 129. Example 13 Solving an Equation Involving a Rational Exponent 共x ⫺ 4兲2兾3 ⫽ 25 3 共x ⫺ 4兲2 ⫽ 25 冪 共x ⫺ 4兲 ⫽ 15,625 2 x ⫺ 4 ⫽ ± 125 x ⫽ 129, x ⫽ ⫺121 Now try Exercise 137. Original equation Rewrite in radical form. Cube each side. Take square root of each side. Add 4 to each side. A58 Appendix A Review of Fundamental Concepts of Algebra Equations with Absolute Values To solve an equation involving an absolute value, remember that the expression inside the absolute value signs can be positive or negative. This results in two separate equations, each of which must be solved. For instance, the equation ⱍx ⫺ 2ⱍ ⫽ 3 results in the two equations x ⫺ 2 ⫽ 3 and ⫺ 共x ⫺ 2兲 ⫽ 3, which implies that the equation has two solutions: x ⫽ 5 and x ⫽ ⫺1. Example 14 ⱍ Solving an Equation Involving Absolute Value ⱍ Solve x 2 ⫺ 3x ⫽ ⫺4x ⫹ 6. Solution Because the variable expression inside the absolute value signs can be positive or negative, you must solve the following two equations. First Equation x 2 ⫺ 3x ⫽ ⫺4x ⫹ 6 Use positive expression. x ⫹x⫺6⫽0 2 Write in general form. 共x ⫹ 3兲共x ⫺ 2兲 ⫽ 0 Factor. x⫹3⫽0 x ⫽ ⫺3 Set 1st factor equal to 0. x⫺2⫽0 x⫽2 Set 2nd factor equal to 0. Second Equation ⫺ 共x 2 ⫺ 3x兲 ⫽ ⫺4x ⫹ 6 Use negative expression. x 2 ⫺ 7x ⫹ 6 ⫽ 0 Write in general form. 共x ⫺ 1兲共x ⫺ 6兲 ⫽ 0 Factor. x⫺1⫽0 x⫽1 Set 1st factor equal to 0. x⫺6⫽0 x⫽6 Set 2nd factor equal to 0. Check ? ⱍ共⫺3兲2 ⫺ 3共⫺3兲ⱍ ⫽ ⫺4共⫺3兲 ⫹ 6 Substitute ⫺3 for x. ⱍ Substitute 2 for x. ✓ 18 ⫽ 18 ? 共2兲2 ⫺ 3共2兲 ⫽ ⫺4共2兲 ⫹ 6 ⫺3 checks. 2 ⫽ ⫺2 ? 共1兲2 ⫺ 3共1兲 ⫽ ⫺4共1兲 ⫹ 6 2 does not check. 2⫽2 ? 共6兲2 ⫺ 3共6兲 ⫽ ⫺4共6兲 ⫹ 6 1 checks. ⱍ ⱍ ⱍ ⱍ ⱍ 18 ⫽ ⫺18 The solutions are x ⫽ ⫺3 and x ⫽ 1. Now try Exercise 151. Substitute 1 for x. ✓ Substitute 6 for x. 6 does not check. Appendix A.5 Solving Equations A59 Common Formulas The following geometric formulas are used at various times throughout this course. For your convenience, some of these formulas along with several others are also provided on the inside cover of this text. Common Formulas for Area A, Perimeter P, Circumference C, and Volume V Rectangle Circle Triangle Rectangular Solid Circular Cylinder Sphere A ⫽ lw A ⫽ r2 V ⫽ lwh V ⫽ r 2h 4 V ⫽ r3 3 P ⫽ 2l ⫹ 2w C ⫽ 2 r 1 A ⫽ bh 2 P⫽a⫹b⫹c w c a h r w l h Using a Geometric Formula A cylindrical can has a volume of 200 cubic centimeters 共cm3兲 and a radius of 4 centimeters (cm), as shown in Figure A.13. Find the height of the can. 4 cm h A.13 r l b Example 15 FIGURE r h Solution The formula for the volume of a cylinder is V ⫽ r 2h. To find the height of the can, solve for h. h⫽ V r2 Then, using V ⫽ 200 and r ⫽ 4, find the height. h⫽ ⫽ 200 共4兲 2 Substitute 200 for V and 4 for r. 200 16 Simplify denominator. ⬇ 3.98 Use a calculator. Because the value of h was rounded in the solution, a check of the solution will not result in an equality. If the solution is valid, the expressions on each side of the equal sign will be approximately equal to each other. V ⫽ r2 h ? 200 ⬇ 共4兲2共3.98兲 200 ⬇ 200.06 Write original equation. Substitute 200 for V, 4 for r, and 3.98 for h. Solution checks. ✓ You can also use unit analysis to check that your answer is reasonable. 200 cm3 ⬇ 3.98 cm 16 cm2 Now try Exercise 157. A60 Appendix A A.5 Review of Fundamental Concepts of Algebra EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. 1. A(n) ________ is a statement that equates two algebraic expressions. 2. A linear equation in one variable is an equation that can be written in the standard form ________. 3. When solving an equation, it is possible to introduce an ________ solution, which is a value that does not satisfy the original equation. 4. The four methods that can be used to solve a quadratic equation are ________, ________, ________, and the ________. SKILLS AND APPLICATIONS In Exercises 5–12, determine whether the equation is an identity or a conditional equation. 5. 6. 7. 8. 9. 10. 4共x ⫹ 1兲 ⫽ 4x ⫹ 4 2共x ⫺ 3兲 ⫽ 7x ⫺ 1 ⫺6共x ⫺ 3兲 ⫹ 5 ⫽ ⫺2x ⫹ 10 3共x ⫹ 2兲 ⫺ 5 ⫽ 3x ⫹ 1 36. 4共x ⫹ 1兲 ⫺ 2x ⫽ 2共x ⫹ 2兲 x 2 ⫹ 2共3x ⫺ 2兲 ⫽ x 2 ⫹ 6x ⫺ 4 11. 3 ⫹ 1 4x ⫽ x⫹1 x⫹1 12. 37. 5 3 ⫹ ⫽ 24 x x In Exercises 13–26, solve the equation and check your solution. 13. 15. 17. 19. 20. 21. 22. x ⫹ 11 ⫽ 15 7 ⫺ 2x ⫽ 25 8x ⫺ 5 ⫽ 3x ⫹ 20 4y ⫹ 2 ⫺ 5y ⫽ 7 ⫺ 6y 3共x ⫹ 3兲 ⫽ 5共1 ⫺ x兲 ⫺ 1 14. 7 ⫺ x ⫽ 19 16. 7x ⫹ 2 ⫽ 23 18. 7x ⫹ 3 ⫽ 3x ⫺ 17 In Exercises 27–42, solve the equation and check your solution. (If not possible, explain why.) 27. x ⫹ 8 ⫽ 2共x ⫺ 2兲 ⫺ x 28. 8共x ⫹ 2兲 ⫺ 3共2x ⫹ 1兲 ⫽ 2共x ⫹ 5兲 100 ⫺ 4x 5x ⫹ 6 29. ⫽ ⫹6 3 4 17 ⫹ y 32 ⫹ y 30. ⫹ ⫽ 100 y y 5x ⫺ 4 2 ⫽ 5x ⫹ 4 3 39. 41. 42. 1 2 2 ⫽ ⫹ 共x ⫺ 4兲共x ⫺ 2兲 x ⫺ 4 x ⫺ 2 1 3 4 ⫹ ⫽ 2 x⫺2 x⫹3 x ⫹x⫺6 3 4 1 6 2 3共x ⫹ 5兲 40. ⫺ ⫹ ⫽ ⫽ 2 2 x ⫺ 3x x x⫺3 x x⫹3 x ⫹ 3x 2 2 共x ⫹ 2兲 ⫹ 5 ⫽ 共x ⫹ 3兲 共2x ⫹ 1兲2 ⫽ 4共x 2 ⫹ x ⫹ 1兲 In Exercises 43– 46, write the quadratic equation in general form. 44. 13 ⫺ 3共x ⫹ 7兲2 ⫽ 0 46. x共x ⫹ 2兲 ⫽ 5x 2 ⫹ 1 In Exercises 47– 58, solve the quadratic equation by factoring. x 3x x 24. ⫺ ⫽ 3 ⫹ 5 2 10 25. 32共z ⫹ 5兲 ⫺ 14共z ⫹ 24兲 ⫽ 0 26. 0.60x ⫹ 0.40共100 ⫺ x兲 ⫽ 50 31. 38. 8x 7 ⫺ ⫽ ⫺4 2x ⫹ 1 2x ⫺ 1 43. 2x 2 ⫽ 3 ⫺ 8x 45. 15共3x 2 ⫺ 10兲 ⫽ 18x x ⫺ 3共2x ⫹ 3兲 ⫽ 8 ⫺ 5x 9x ⫺ 10 ⫽ 5x ⫹ 2共2x ⫺ 5兲 3x 4x 23. ⫺ ⫽4 8 3 2 1 2 34. ⫹ ⫽0 z⫹2 x x⫺5 x 4 35. ⫹ ⫹2⫽0 x⫹4 x⫹4 33. 3 ⫽ 2 ⫹ 32. 6 15 ⫺4⫽ ⫹3 x x 47. 49. 51. 53. 55. 57. 58. 6x 2 ⫹ 3x ⫽ 0 x 2 ⫺ 2x ⫺ 8 ⫽ 0 x2 ⫺ 12x ⫹ 35 ⫽ 0 3 ⫹ 5x ⫺ 2x 2 ⫽ 0 48. 9x 2 ⫺ 4 ⫽ 0 50. x 2 ⫺ 10x ⫹ 9 ⫽ 0 52. 4x 2 ⫹ 12x ⫹ 9 ⫽ 0 54. 2x 2 ⫽ 19x ⫹ 33 2 56. 18 x 2 ⫺ x ⫺ 16 ⫽ 0 x ⫹ 4x ⫽ 12 x 2 ⫹ 2ax ⫹ a 2 ⫽ 0, a is a real number 共x ⫹ a兲2 ⫺ b 2 ⫽ 0, a and b are real numbers In Exercises 59–70, solve the equation by extracting square roots. 59. 61. 63. 65. 67. 69. x 2 ⫽ 49 3x 2 ⫽ 81 共x ⫺ 12兲2 ⫽ 16 共x ⫹ 2兲 2 ⫽ 14 共2x ⫺ 1兲2 ⫽ 18 共x ⫺ 7兲2 ⫽ 共x ⫹ 3兲 2 60. 62. 64. 66. 68. 70. x 2 ⫽ 32 9x 2 ⫽ 36 共x ⫹ 13兲2 ⫽ 25 共x ⫺ 5兲2 ⫽ 30 共2x ⫹ 3兲2 ⫺ 27 ⫽ 0 共x ⫹ 5兲2 ⫽ 共x ⫹ 4兲 2 Appendix A.5 In Exercises 71– 80, solve the quadratic equation by completing the square. 71. 73. 75. 77. 79. x2 x2 ⫹ 4x ⫺ 32 ⫽ 0 ⫹ 12x ⫹ 25 ⫽ 0 8 ⫹ 4x ⫺ x 2 ⫽ 0 2x 2 ⫹ 5x ⫺ 8 ⫽ 0 5x2 ⫺ 15x ⫹ 7 ⫽ 0 72. 74. 76. 78. 80. ⫹ 6x ⫹ 2 ⫽ 0 x ⫹ 8x ⫹ 14 ⫽ 0 9x 2 ⫺ 12x ⫽ 14 4x 2 ⫺ 4x ⫺ 99 ⫽ 0 3x2 ⫹ 9x ⫹ 5 ⫽ 0 x2 2 In Exercises 81–98, use the Quadratic Formula to solve the equation. 81. 83. 85. 87. 89. 91. 93. 95. 97. 2x 2 ⫹ x ⫺ 1 ⫽ 0 2 ⫹ 2x ⫺ x 2 ⫽ 0 x 2 ⫹ 14x ⫹ 44 ⫽ 0 x 2 ⫹ 8x ⫺ 4 ⫽ 0 12x ⫺ 9x 2 ⫽ ⫺3 9x2 ⫹ 24x ⫹ 16 ⫽ 0 28x ⫺ 49x 2 ⫽ 4 8t ⫽ 5 ⫹ 2t 2 共 y ⫺ 5兲2 ⫽ 2y 82. 84. 86. 88. 90. 92. 94. 96. 98. 25x 2 ⫺ 20x ⫹ 3 ⫽ 0 x 2 ⫺ 10x ⫹ 22 ⫽ 0 6x ⫽ 4 ⫺ x 2 4x 2 ⫺ 4x ⫺ 4 ⫽ 0 16x 2 ⫹ 22 ⫽ 40x 16x 2 ⫺ 40x ⫹ 5 ⫽ 0 3x ⫹ x 2 ⫺ 1 ⫽ 0 25h2 ⫹ 80h ⫹ 61 ⫽ 0 共57x ⫺ 14兲2 ⫽ 8x In Exercises 99–104, use the Quadratic Formula to solve the equation. (Round your answer to three decimal places.) 99. 100. 101. 102. 103. 104. 0.1x2 ⫹ 0.2x ⫺ 0.5 ⫽ 0 2x 2 ⫺ 2.50x ⫺ 0.42 ⫽ 0 ⫺0.067x 2 ⫺ 0.852x ⫹ 1.277 ⫽ 0 ⫺0.005x 2 ⫹ 0.101x ⫺ 0.193 ⫽ 0 422x 2 ⫺ 506x ⫺ 347 ⫽ 0 ⫺3.22x 2 ⫺ 0.08x ⫹ 28.651 ⫽ 0 In Exercises 105–112, solve the equation using any convenient method. 105. 107. 109. 111. 112. 106. 11x ⫹ 33x ⫽ 0 108. x2 ⫺ 14x ⫹ 49 ⫽ 0 共x ⫹ 3兲 ⫽ 81 11 110. 3x ⫹ 4 ⫽ 2x2 ⫺ 7 x2 ⫺ x ⫺ 4 ⫽ 0 4x 2 ⫹ 2x ⫹ 4 ⫽ 2x ⫹ 8 a 2x 2 ⫺ b 2 ⫽ 0, a and b are real numbers, a ⫽ 0 x2 ⫺ 2x ⫺ 1 ⫽ 0 2 2 In Exercises 113–126, find all real solutions of the equation. Check your solutions in the original equation. 113. 115. 117. 118. 119. 120. 114. 20x3 ⫺ 125x ⫽ 0 2x4 ⫺ 50x2 ⫽ 0 116. x6 ⫺ 64 ⫽ 0 x 4 ⫺ 81 ⫽ 0 3 x ⫹ 216 ⫽ 0 9x4 ⫺ 24x3 ⫹ 16x 2 ⫽ 0 x3 ⫺ 3x2 ⫺ x ⫹ 3 ⫽ 0 x3 ⫹ 2x2 ⫹ 3x ⫹ 6 ⫽ 0 121. 122. 123. 125. Solving Equations A61 x4 ⫹ x ⫽ x3 ⫹ 1 x4 ⫺ 2x3 ⫽ 16 ⫹ 8x ⫺ 4x3 124. 36t 4 ⫹ 29t 2 ⫺ 7 ⫽ 0 x4 ⫺ 4x2 ⫹ 3 ⫽ 0 126. x6 ⫹ 3x3 ⫹ 2 ⫽ 0 x6 ⫹ 7x3 ⫺ 8 ⫽ 0 In Exercises 127–154, find all solutions of the equation. Check your solutions in the original equation. 127. 冪2x ⫺ 10 ⫽ 0 128. 7冪x ⫺ 6 ⫽ 0 129. 冪x ⫺ 10 ⫺ 4 ⫽ 0 130. 冪5 ⫺ x ⫺ 3 ⫽ 0 131. 冪2x ⫹ 5 ⫹ 3 ⫽ 0 132. 冪3 ⫺ 2x ⫺ 2 ⫽ 0 3 3 4x ⫺ 3 ⫹ 2 ⫽ 0 133. 冪2x ⫹ 1 ⫹ 8 ⫽ 0 134. 冪 135. 冪5x ⫺ 26 ⫹ 4 ⫽ x 136. 冪x ⫹ 5 ⫽ 冪2x ⫺ 5 3兾2 137. 共x ⫺ 6兲 ⫽ 8 138. 共x ⫹ 3兲3兾2 ⫽ 8 139. 共x ⫹ 3兲2兾3 ⫽ 5 140. 共x2 ⫺ x ⫺ 22兲4兾3 ⫽ 16 141. 3x共x ⫺ 1兲1兾2 ⫹ 2共x ⫺ 1兲3兾2 ⫽ 0 142. 4x2共x ⫺ 1兲1兾3 ⫹ 6x共x ⫺ 1兲4兾3 ⫽ 0 143. x ⫽ 145. 3 1 ⫹ x 2 144. 20 ⫺ x ⫽x x 3 4 ⫺ ⫽1 x⫹1 x⫹2 146. 4x ⫹ 1 ⫽ x 1 ⫹ ⫽3 x2 ⫺ 4 x ⫹ 2 149. 2x ⫺ 1 ⫽ 5 151. x ⫽ x 2 ⫹ x ⫺ 3 153. x ⫹ 1 ⫽ x 2 ⫺ 5 147. ⱍ ⱍ ⱍⱍ ⱍ ⱍ 3 x x⫹1 x⫹1 ⫺ ⫽0 3 x⫹2 150. 13x ⫹ 1 ⫽ 12 152. x 2 ⫹ 6x ⫽ 3x ⫹ 18 154. x ⫺ 10 ⫽ x 2 ⫺ 10x 148. ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ANTHROPOLOGY In Exercises 155 and 156, use the following information. The relationship between the length of an adult’s femur (thigh bone) and the height of the adult can be approximated by the linear equations y ⴝ 0.432 x ⴚ 10.44 Female y ⴝ 0.449x ⴚ 12.15 Male where y is the length of the femur in inches and x is the height of the adult in inches (see figure). x in. y in. femur 155. An anthropologist discovers a femur belonging to an adult human female. The bone is 16 inches long. Estimate the height of the female. A62 Appendix A Review of Fundamental Concepts of Algebra 156. From the foot bones of an adult human male, an anthropologist estimates that the person’s height was 69 inches. A few feet away from the site where the foot bones were discovered, the anthropologist discovers a male adult femur that is 19 inches long. Is it likely that both the foot bones and the thigh bone came from the same person? 157. VOLUME OF A BILLIARD BALL A billiard ball has a volume of 5.96 cubic inches. Find the radius of a billiard ball. 158. LENGTH OF A TANK The diameter of a cylindrical propane gas tank is 4 feet. The total volume of the tank is 603.2 cubic feet. Find the length of the tank. 159. GEOMETRY A “Slow Moving Vehicle” sign has the shape of an equilateral triangle. The sign has a perimeter of 129 centimeters. Find the length of each side of the sign. Find the area of the sign. 160. GEOMETRY The radius of a traffic cone is 14 centimeters and the lateral surface area of the cone is 1617 square centimeters. Find the height of the cone. 161. VOTING POPULATION The total voting-age population P (in millions) in the United States from 1990 through 2006 can be modeled by P⫽ 182.17 ⫺ 1.542t , 0 ⱕ t ⱕ 16 1 ⫺ 0.018t where t represents the year, with t ⫽ 0 corresponding to 1990. (Source: U.S. Census Bureau) (a) In which year did the total voting-age population reach 200 million? (b) Use the model to predict the year in which the total voting-age population will reach 241 million. Is this prediction reasonable? Explain. 162. AIRLINE PASSENGERS An airline offers daily flights between Chicago and Denver. The total monthly cost C (in millions of dollars) of these flights is C ⫽ 冪0.2x ⫹ 1, where x is the number of passengers (in thousands). The total cost of the flights for June is 2.5 million dollars. How many passengers flew in June? EXPLORATION TRUE OR FALSE? In Exercises 163 and 164, determine whether the statement is true or false. Justify your answer. 163. An equation can never have more than one extraneous solution. 164. When solving an absolute value equation, you will always have to check more than one solution. 165. THINK ABOUT IT What is meant by equivalent equations? Give an example of two equivalent equations. 166. Solve 3共x ⫹ 4兲2 ⫹ 共x ⫹ 4兲 ⫺ 2 ⫽ 0 in two ways. (a) Let u ⫽ x ⫹ 4, and solve the resulting equation for u. Then solve the u-solution for x. (b) Expand and collect like terms in the equation, and solve the resulting equation for x. (c) Which method is easier? Explain. THINK ABOUT IT In Exercises 167–170, write a quadratic equation that has the given solutions. (There are many correct answers.) ⫺3 and 6 ⫺4 and ⫺11 167. 168. 169. 170. 1 ⫹ 冪2 and 1 ⫺ 冪2 ⫺3 ⫹ 冪5 and ⫺3 ⫺ 冪5 In Exercises 171 and 172, consider an equation of the form x ⴙ x ⴚ a ⴝ b, where a and b are constants. ⱍ ⱍ 171. Find a and b when the solution of the equation is x ⫽ 9. (There are many correct answers.) 172. WRITING Write a short paragraph listing the steps required to solve this equation involving absolute values, and explain why it is important to check your solutions. In Exercises 173 and 174, consider an equation of the form x ⴙ 冪x ⴚ a ⴝ b, where a and b are constants. 173. Find a and b when the solution of the equation is x ⫽ 20. (There are many correct answers.) 174. WRITING Write a short paragraph listing the steps required to solve this equation involving radicals, and explain why it is important to check your solutions. 175. Solve each equation, given that a and b are not zero. (a) ax 2 ⫹ bx ⫽ 0 (b) ax 2 ⫺ ax ⫽ 0 176. CAPSTONE (a) Explain the difference between a conditional equation and an identity. (b) Give an example of an absolute value equation that has only one solution. (c) State the Quadratic Formula in words. (d) Does raising each side of an equation to the nth power always yield an equivalent equation? Explain.