Completing the Square

Completing The Square 1.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 2.Take the square root of both sides 3.The numerical side gets a plus and minus 4.Simplify the variable side

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

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

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

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

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

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

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

Example 3-3a Solveby completing the square. Step 1Isolate the x 2 and x terms. Original equation Subtract 5 from each side. Simplify.

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

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

Example 3-3d CheckSubstitute each value for x in the original equation. Answer: The solution set is {1, 17}.

Solving a problem by completing the square Arrange terms as follows x 2 + 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

Example 3-3e Answer: {–2, 10} Solve

Answer: {–5, 2} Solve

When a ≠ 1 Divide every term by “a”, so that “a” does equal one. First step becomes Arrange terms as follows x 2 + (b/a) x = (-c/a)

Homework 10-3 Completing the Square Two Pages First Column

Example 3-4a Boating Suppose the rate of flow of an 80-foot-wide river is given by the equation where r is the rate in miles per hour, and x is the distance from the shore in feet. Joacquim does not want to paddle his canoe against a current faster than 5 miles per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour? ExploreYou 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.

Example 3-4b PlanFind the distance whenUse completing the square to solve Solve Equation for the current Divide each side by –0.01. Simplify.

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

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

Example 3-4e ExamineThe 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, Answer: He must stay within about 7 feet of either bank.

Boating Suppose the rate of flow of a 6-foot-wide river is given by the equation where r is the rate in miles per hour, and x is the distance from the shore in feet. Joacquim does not want to paddle his canoe against a current faster than 5 files per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour. Example 3-4f Answer: He must stay within 10 feet of either bank.