1. Copyright © Cengage Learning. All rights reserved. Roots, Radical Expressions, and Radical Equations 8

2. Copyright © Cengage Learning. All rights reserved. Section 8.6 Solving Equations Containing Radicals; the Distance Formula

3. 3 Objectives 1.Solve a radical equation containing one square root term. Solve a radical equation containing two square root terms. Solve a radical equation containing a cube root. Find the distance between two points. Solve an application containing a radical equation. 1 1 2 2 3 3 4 4 5 5

4. 4 Solve a radical equation containing one square root term 1.

5. 5 Solve a radical equation containing one square root term We have a power rule to help us solve radical equations. Power Rule for Radicals If a = b, then a n = b n. This states that if both sides of an equation are raised to the same power, the solutions of the original equation will be part of the solutions of the new equation. However, there could be extraneous solutions among the solutions of the new equation.

6. 6 Solve a radical equation containing one square root term Before solving equations containing radicals, we note that if two numbers are equal, their squares are equal. In particular, the squaring property of equality states: If a = b, then a 2 = b 2. If we square both sides of an equation, the resulting equation may or may not have the same solutions as the original one.

7. 7 Solve a radical equation containing one square root term For example, if we square both sides of the equation (1)x = 2, with the solution 2, we obtain x 2 = 2 2, which simplifies as (2)x 2 = 4, with solutions of 2 and –2. Equations 1 and 2 are not equivalent, because they have different solution sets. The solution –2 of Equation 2 does not satisfy Equation 1. Because squaring both sides of an equation can produce an extraneous solution, we must always check each possible solution in the original equation. 2 2 = 4 and (–2) 2 = 4.

8. 8 Solve a radical equation containing one square root term To solve an equation containing square roots, we follow these steps. Solving Equations Containing Radicals 1. Whenever possible, isolate a single radical term on one side of the equation. 2. Raise each side of the equation to a power equal to the index. 3. If the equation still contains a radical term, repeat steps 1 and 2.

9. 9 Solve a radical equation containing one square root term 4. Check the possible solutions in the original equation for extraneous solutions.

10. 10 Solve: Solution: To solve the equation, we note that the radical is already isolated on one side. We proceed to Step 2 and square both sides to eliminate the radical. Since this might produce an extraneous solution, we must check each solution. Example

11. 11 Square both sides of the equation. Subtract 2 from both sides. cont’d Example – Solution

12. 12 We check by substituting 7 for x in the original equation. The solution checks. Since no solutions are lost in this process, 7 is the only solution of the original equation. Substitute 7 for x. cont’d Example – Solution

13. 13 Solve a radical equation containing two square root term 2.

14. 14 Example Solve: Solution: We square both sides to eliminate the radicals. Square both sides of the equation. Simplify.

15. 15 Example – Solution x + 12 = 9x + 36 –8x = 24 x = –3 We check the solution by substituting –3 for x in the original equation. The value –3 checks and is the only solution. Distribute. Subtract 9x and 12 from both sides. Divide both sides by –8. Substitute –3 for x. cont’d

16. 16 Solve a radical equation containing a cube root 3.

17. 17 Solve a radical equation containing a cube root The power rule for radicals can be applied to cube roots. If a = b, then a 3 = b 3. To solve an equation involving a cube root, we isolate the radical term and then cube both sides of the equation.

18. 18 Example Solve: Solution: We can cube both sides and proceed as follows: 2x + 10 = 8 2x = –2 x = –1 Check the result. Cube both sides. Simplify. Subtract 10 from both sides. Divide both sides by 2.

19. 19 Find the distance between two points 4.

20. 20 Find the distance between two points We can use the Pythagorean theorem to derive a formula for finding the distance between two points P(x 1, y 1 ) and Q(x 2, y 2 ) on a rectangular coordinate system. The distance d between points P and Q is the length of the hypotenuse of the triangle in Figure 8-14. The two legs have lengths x 2 – x 1 and y 2 – y 1. Figure 8-14

21. 21 Find the distance between two points By the Pythagorean theorem, we have d 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 We can take the square root of both sides of the equation to get the distance formula. The Distance Formula The distance d between points P(x 1, y 1 ) and Q(x 2, y 2 ) is given by the formula

22. 22 Example Find the distance between points P(1, 5) and Q(4, 9). (See Figure 8-15.) Figure 8-15

23. 23 Example – Solution We can use the distance formula by substituting 1 for x 1, 5 for y 1, 4 for x 2, and 9 for y 2. The distance between points P and Q is 5 units.

24. 24 Solve an application containing a radical equation 5.

25. 25 Example – Building Freeways In a city, streets run north and south, and avenues run east and west. Consecutive streets are 750 feet apart and consecutive avenues are 750 feet apart. The city plans to construct a freeway from the intersection of 21st Street and 4th Avenue to the intersection of 111th Street and 60th Avenue. How long will it be in miles?

26. 26 Example – Solution We can represent the roads of the city by the coordinate system shown in Figure 8-16, where each unit on each axis represents 750 feet. Figure 8-16

27. 27 Example – Solution We represent the end of the freeway at 21st Street and 4th Avenue by the point (x 1, y 1 ) = (21, 4). The other end is (x 1, y 1 ) = (111, 60). We now can use the distance formula to find the length of the freeway in blocks. cont’d

28. 28 Example – Solution The freeway will be 106 blocks in length. Since each block represents 750 feet, the length of the freeway will be 106  750 = 79,500 feet. Since there are 5,280 feet in 1 mile, we can divide 79,500 by 5,280 to convert 79,500 feet to 15.056818 miles. The freeway will be about 15 miles long. Find the square root. cont’d