5-Minute Check on Chapter 2 Transparency 3-1 5-Minute Check on Chapter 2 Evaluate 42 - |x - 7| if x = -3 Find 4.1  (-0.5) Simplify each expression 3. 8(-2c + 5) + 9c 4. (36d – 18) / (-9) A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops. If one is chosen at random, what is the probability that it is not green? Which of the following is a true statement Standardized Test Practice: A 8/4 < 4/8 B -4/8 < -8/4 C -4/8 > -8/4 D -4/8 > 4/8 Click the mouse button or press the Space Bar to display the answers.

Solving Quadratic Equations by Completing the Square Lesson 10-3 Solving Quadratic Equations by Completing the Square

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Objectives Solve quadratic equations by finding the square root Solve quadratic equations by completing the square

Vocabulary Completing the square –

Multiplication and Division PoE Properties of Equality (PoE) are based on the concept that as long as you do the same thing to both sides of an equation, then you have not changed anything. Multiplication PoE For any numbers a, b, and c, if a = b, then ac = bc You can multiply both sides of an equation by the same thing without changing the equation Division PoE For any numbers a, b, and c with c ≠ 0, if a = b, then a/c = b/c You can divide both sides of an equation by the same thing without changing the equation Multiplication and division are reciprocal actions

Example 1 Solve by taking the square root of each side. Round to the nearest tenth if necessary. is a perfect square trinomial. Take the square root of each side. Simplify. Definition of absolute value Subtract 3 from each side. Simplify. Use a calculator to evaluate each value of x. or Answer: The solution set is {–5.2, –0.8}.

Example 2 Find the value of c that makes a perfect square. Method 1 Use algebra tiles. is a perfect square.

Example 2 cont Method 2 Complete the square. 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 Solve by completing the square. Step 1 Isolate the x2 and x terms. Original equation Subtract 5 from each side Simplify. Step 2 Complete the square and solve. Since , add 81 to each side. Factor

Example 3 cont Factor Take the square root of each side. Add 9 to each side. Simplify. or Answer: The solution set is {1, 17}.

Example 4 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? 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.

Example 4 cont Plan Find the distance when Use completing the square to solve Solve Equation for the current Divide each side by –0.01. Simplify. Since ½(-80)² = 1600 add 1600 to each side. Factor

Example 4 cont Take the square root of each side. Add 40 to each side Simplify. Use a calculator to evaluate each value of x. or 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, Answer: He must stay within about 7 feet of either bank.

Summary & Homework Summary: Homework: Complete the square to make a quadratic function a perfect square Use the following steps to complete the square of x2 + bx: Step 1. Find ½ of b, the coefficient of x Step 2. Square the result of step 1 Step 3. Add the result of step 2 to both sides of the equation Homework: pg