Completing the Square

1. Completing the Square

2. Completing The Square • Make the quadratic equation on one side of the equal sign into a perfect square • Add to both sides to make the last term correct • Take the square root of both sides • The numerical side gets a plus and minus • Simplify the variable side

3. Solve by taking the square root of each side. Round to the nearest tenth if necessary. Definition of absolute value Original equation is a perfect square trinomial. Take the square root of each side. Simplify. Example 3-1a

4. Subtract 3 from each side. Simplify. or Example 3-1b Use a calculator to evaluate each value of x. Answer: The solution set is {–5.2, –0.8}.

5. Solve by taking the square root of each side. Round to the nearest tenth if necessary. Example 3-1c Answer: {–2.3, –5.7}

6. Find the value of c that makes a perfect square. Example 3-2a

7. Step 1 Find Step 2 Square the result of Step 1. Step 3 Add the result of Step 2 to Answer: Notice that Example 3-2b Complete the square.

8. Find the value of c that makes a perfect square. Answer: Example 3-2c

9. Perfect Square Process The last term is one-half the middle term squared e.g. x2 + 10x The last term should be (½ * 10)2 = 25

10. Solve by completing the square. Original equation Subtract 5 fromeach side. Simplify. Example 3-3a Step 1 Isolate the x2 and x terms.

11. Since ,add 81 to each side. Factor Take the square root of each side. Example 3-3b Step 2 Complete the square and solve.

12. Add 9 to each side. Simplify. or Example 3-3c

13. Example 3-3d Check Substitute each value for x in the originalequation. Answer: The solution set is {1, 17}.

14. Solving a problem by completing the square • Arrange terms as follows x2 + bx = -c • Complete the square, adding the same constant to both sides of the equation. (The last term is one-half the middle term squared) • Square root of both sides • Solve for x, there can be up to two answers

15. Solve Example 3-3e Answer: {–2, 10}

16. Solve Answer: {–5, 2}

17. When a ≠ 1 • Divide every term by “a”, so that “a” does equal one. • First step becomes Arrange terms as follows x2 + (b/a) x = (-c/a)

18. Homework • 10-3 Completing the Square Two Pages First Column

19. Boating Suppose the rate of flowof an 80-foot-wide river isgiven by the equationwhere r is the rate in miles per hour, and x is the distance from the shore in feet.Joacquim does not want to paddle hiscanoe against a current faster than 5 miles per hour.At what distance from the river bank must hepaddle in order to avoid a current of 5 miles perhour? Example 3-4a Explore You know the function that relates distance from shore to the rate of the river current. You want to know how far away from the river bank he must paddle to avoid the current.

20. Plan Find the distance when Use completing the square to solve Equation for the current Solve Divide each side by –0.01. Simplify. Example 3-4b

21. Sinceadd 1600 to each side. Factor Take the square root of each side. Example 3-4c

22. Add 40 to each side. Simplify. or Example 3-4d Use a calculator to evaluate each value of x.

23. Examine The solutions of the equation are about 7 ft and about 73 ft. The solutions are distances from one shore. Since the river is about 80 ft wide, Example 3-4e Answer: He must stay within about 7 feet of either bank.

24. BoatingSuppose the rate of flow of a 6-foot-wide river is givenby the equationwhere r is the rate in miles per hour, and x is the distance from the shore in feet. Joacquimdoes not want to paddlehis canoe against a current faster than 5 files per hour.At what distance from the river bank must he paddlein order to avoid a current of 5 miles perhour. Example 3-4f Answer: He must stay within 10 feet of either bank.