College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson

Equations and Inequalities 1

Other Types of Equations 1.5

Other Types of Equations So far, we learned how to solve linear and quadratic equations. In this section, we study other types of equations, including those that involve: Higher powers Fractional expressions Radicals

Polynomial Equations

Some equations can be solved by factoring and using the Zero-Product Property. The Zero-Product Property says that if a product equals 0, then at least one of the factors must equal 0.

E.g. 1—Solving an Equation by Factoring Solve the equation x 5 = 9x 3 We bring all terms to one side and then factor.

E.g. 1—Solving an Equation by Factoring x 5 = 9x 3 x 3 (x 2 – 9) = 0 x 3 (x – 3)(x + 3) = 0 x 3 = 0 or x – 3 = 0 or x + 3 = 0 x = 0 x = 3 x = –3 You should check that each of these satisfies the original equation.

Caution To divide each side of the equation in Example 1 by the common factor x 3 would be wrong. In doing so, we would lose the solution x = 0. Never divide both sides of an equation by an expression that contains the variable unless you know that the expression cannot equal 0.

E.g. 2—Factoring by Grouping Solve the equation x 3 + 3x 2 – 4x – 12 = 0 The left-hand side of the equation can be factored by grouping the terms in pairs.

E.g. 2—Factoring by Grouping (x 3 + 3x 2) – (4x + 12) = 0 x 2 (x + 3) – 4(x + 3) = 0 (x 2 – 4)(x + 3) = 0 (x – 2)(x + 2)(x + 3) = 0 x – 2 = 0 or x + 2 = 0 or x + 3 = 0 x = 0 x = –2 x = –3

E.g. 3—An Equation Involving Fractional Expressions Solve the equation To simplify the equation, we multiplying each side by the lowest common denominator (LCD).

E.g. 3—An Equation Involving Fractional Expressions

We must check our answers because multiplying by an expression that contains the variable can introduce extraneous solutions.

E.g. 3—An Equation Involving Fractional Expressions We see that the solutions are x = 3 and –1.

Equations Involving Radicals

When you solve an equation that involves radicals, you must be especially careful to check your final answers. The next example demonstrates why.

E.g. 4—An Equation Involving a Radical Solve the equation To eliminate the square root, we first isolate it on one side of the equal sign, then square.

E.g. 4—An Equation Involving a Radical

The values x = –¼ and x = 1 are only potential solutions. We must check them to see if they satisfy the original equation.

E.g. 4—An Equation Involving a Radical We see x = –¼ is a solution but x = 1 is not.

Extraneous Solutions When we solve an equation, we may end up with one or more extraneous solutions— potential solutions that do not satisfy the original equation. In Example 4, the value x = 1 is an extraneous solution.

Extraneous Solutions Extraneous solutions may be introduced when we square each side of an equation because the operation of squaring can turn a false equation into a true one. For example, –1 ≠ 1 but (–1) 2 = 1 2

Extraneous Solutions Thus, the squared equation may be true for more values of the variable than the original equation. That is why you must always check your answers to make sure each satisfies the original equation.

Equations of Quadratic Type

Quadratic Type Equation An equation of the form aW 2 + bW + c = 0, where W is an algebraic expression, is an equation of quadratic type. We solve these equations by substituting for the algebraic expression—as we see in the next two examples.

E.g. 5—An Equation of Quadratic Type Find all solutions of: We could solve this equation by multiplying it out first. But it’s easier to think of the expression 1 + 1/x as the unknown and give it a new name W.

E.g. 5—An Equation of Quadratic Type This turns the equation into a quadratic equation in the new variable W.

E.g. 5—An Equation of Quadratic Type Now we change these values of W back into the corresponding values of x.

E.g. 6—A Fourth-Degree Equation of Quadratic Type Find all solutions of: x 4 – 8x = 0 If we set W = x 2, we get a quadratic equation in the new variable W.

E.g. 6—A Fourth-Degree Equation of Quadratic Type (x 2 ) 2 – 8x = 0 (Write x 4 as (x 2 ) 2 ) W 2 – 8W + 8 = 0 (Let W = x 2 )

E.g. 6—A Fourth-Degree Equation of Quadratic Type So, there are four solutions: Using a calculator, we obtain the approximations: x ≈ 2.61, 1.08, –2.61, –1.08

E.g. 7—An Equation Involving Fractional Powers Find all solutions of: x 1/3 + x 1/6 – 2 = 0 This equation is of quadratic type because, if we let W = x 1/6, then W 2 = (x 1/6 ) 2 = x 1/3

E.g. 7—An Equation Involving Fractional Powers x 1/3 + x 1/6 – 2 = 0 W 2 + W – 2 = 0 (Let W = x 1/6 ) (W – 1)(W + 2) = 0 (Factor) W – 1 = 0 or W + 2 = 0 (Zero-Product Property)

E.g. 7—An Equation Involving Fractional Powers W = 1 W = –2 (Solve) x 1/6 = 1 x 1/6 = –2 (W = x 1/6 ) x = 1 6 =1 x = (–2) 6 = 64 (Take the 6th power)

E.g. 7—An Equation Involving Fractional Powers By checking, we see that x = 1 is a solution but x = 64 is not.

Applications

Many real-life problems can be modeled with the types of equations that we have studied in this section.

E.g. 8—Dividing a Lottery Jackpot A group of people come forward to claim a $1,000,000 lottery jackpot. The winners are to share equally.

E.g. 8—Dividing a Lottery Jackpot Before the jackpot is divided, three more winning ticket holders show up. As a result, each person’s share is reduced by $75,000. How many winners were in the original group?

E.g. 8—Dividing a Lottery Jackpot We are asked for the number of people in the original group. So, let: x = number of winners in the original group

E.g. 8—Dividing a Lottery Jackpot In WordsIn Algebra Number of winners in original groupx Number of winners in final groupx + 3 Winnings per person, originally Winnings per person, finally We translate the information in the problem as follows.

E.g. 8—Dividing a Lottery Jackpot Now, we set up the model. Winnings per person, originally – 75,000 = Winnings per person, finally

E.g. 8—Dividing a Lottery Jackpot

Since we can’t have a negative number of people, we conclude that there were five winners in the original group.

E.g. 9—Energy Expended in Bird Flight Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. Air generally rises over land and falls over water in the daytime. So, flying over water requires more energy.

E.g. 9—Energy Expended in Bird Flight A bird is released from point A on an island, 5 mi from B, the nearest point on a straight shoreline. It flies to a point C on the shoreline. Then, it flies along the shoreline to its nesting area D.

E.g. 9—Energy Expended in Bird Flight Suppose it has 170 kcal of energy reserves. It uses 10 kcal/mi flying over land. It uses 14 kcal/mi flying over water.

E.g. 9—Energy Expended in Bird Flight (a)Where should C be located so that the bird uses exactly 170 kcal of energy during its flight? (b)Does it have enough energy reserves to fly directly from A to D?

E.g. 9—Energy in Bird Flight We are asked to find the location of C. So, let: x = distance from B to C Example (a)

E.g. 9—Energy in Bird Flight From the figure, and from the fact that energy used = energy per mile x miles flown we determine the following. Example (a)

E.g. 9—Energy in Bird Flight In WordsIn Algebra Distance from B to Cx Distance flown over water (from A to C) Distance flown over land (from C to D)12 – x Energy used over water Energy used over land10(12 – x) Example (a)

E.g. 9—Energy in Bird Flight Now, we set up the model. Total energy used = Energy used over water + Energy used over land Example (a)

E.g. 9—Energy in Bird Flight Thus, To solve this equation, we eliminate the square root by first bringing all other terms to the left of the equal sign and then squaring each side. Example (a)

E.g. 9—Energy in Bird Flight Example (a)

E.g. 9—Energy in Bird Flight Example (a) This equation could be factored. However, as the numbers are so large, it is easier to use the quadratic formula and a calculator.

E.g. 9—Energy in Bird Flight Example (a) C should be either mi or mi from B so that the bird uses exactly 170 kcal of energy during its flight.

E.g. 9—Energy in Bird Flight By the Pythagorean Theorem, the length of the route directly from A to D is: Example (b)

E.g. 9—Energy in Bird Flight Thus, the energy the bird requires for that route is: 14 x 13 = 182 kcal This is more energy than the bird has available. So, it can’t use this route. Example (b)