Completing the Square

Transcription

1 5-7 Completing the Square TEKS FOCUS TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. Additional TEKS (1)(F), (4)(D) VOCABULARY Completing the square the process of finding a constant c to add to x 2 + bx so that x 2 + bx + c is the square of a binomial Apply use knowledge or information for a specific purpose, such as solving a problem ESSENTIAL UNDERSTANDING Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a binomial. Key Concept Completing the Square You can form a perfect square trinomial from x 2 + bx by adding ( b 2 ) 2. x 2 + bx + ( b 2 ) 2 = (x + b 2 ) 2 1. Rewrite the equation in the form x 2 + bx = c. To do this, get all terms with the variable on one side of the equation and the constant on the other side. Divide all the terms of the equation by the coefficient of x 2 if it is not Complete the square by adding ( b 2 ) 2 to each side of the equation. 3. Factor the trinomial. 4. Find square roots. 5. Solve for x. Key Concept Solving an Equation by Completing the Square 190 Lesson 5-7 Completing the Square

2 Problem 1 Solving by Finding Square Roots How is solving this equation like solving a linear equation? You isolate the variable term. What is the solution of each equation? A 4x = 46 B 3x 2 5 = 25 4x 2 = 36 d Rewrite in ax 2 = c form. S 3x 2 = 30 4x 2 4 = 36 4 d Isolate x 2. S 3x 2 3 = 30 3 x 2 = 9 x 2 = 10 x = {3 d Find square roots. S x = { 110 Problem 2 TEKS Process Standard (1)(A) Determining Dimensions STEM Architecture While designing a house, an architect used windows like the one shown here. What are the dimensions of the window if it has 2766 square inches of glass? Is the answer reasonable? Yes; the rectangular part is about 30 * 70 = 2100 in. 2. This leaves enough glass for the semicircle. Step 1 Find the area of the window. The area of the rectangular part is (2x)(x) = 2x 2 in. 2. The area of the semicircular part is 1 2 pr 2 = 1 2 p ( x 2 ) 2 = 1 2 p x 2 4 = p 8 x 2 in. 2. So, the total amount of glass used is 2x 2 + p 8 x 2 = 2766 in. 2. Step 2 Solve for x. (2 + p 8 ) x 2 = 2766 x 2 = p 8 x {34 Write the equation in ax 2 = c form. Isolate x 2. Find square roots. Use a calculator. Length cannot be negative. So the rectangular portion of the window is 34 in. wide by 68 in. long. The semicircular top has a radius of 17 in. PearsonTEXAS.com 191

3 Problem 3 Solving a Perfect Square Trinomial Equation What is the solution of x 2 + 4x + 4 = 25? Factor the perfect square trinomial. Find square roots. x 2 + 4x + 4 = 25 (x + 2) 2 = 25 x + 2 = t5 Rewrite as two equations. x + 2 = 5 or x + 2 = 5 Solve for x. x = 3 or x = 7 Problem 4 TEKS Process Standard (1)(G) Completing the Square Why do you want a perfect square trinomial? You can factor a perfect square trinomial into the square of a binomial. What value completes the square for x 2 10x? Justify your answer. x 2-10x Identify b; b = -10 ( b 2 ) 2 = ( ) 2 = (-5) 2 = 25 Find ( b 2 ) 2. x 2-10x + 25 x 2-10x + 25 = (x - 5) 2 Add the value of ( b 2 ) 2 to complete the square. Rewrite as the square of a binomial. Problem 5 Solving by Completing the Square What is the solution of 3x 2 12x + 6 = 0? How can you check your work? Use a calculator to estimate the values of x. Then substitute these estimates in the original equation. 3x 2-12x = -6 3x x 3 = x 2-4x = -2 Rewrite. Get all terms with x on one side of the equation. Divide each side by 3 so the coefficient of x 2 will be 1. Simplify. ( b 2 ) 2 = ( ) 2 = (-2) 2 = 4 Find ( b 2 ) 2 = 4. x 2-4x + 4 = Add 4 to each side. (x - 2) 2 = 2 Factor the trinomial. x - 2 = {12 Find square roots. x = 2 { 12 Solve for x. 192 Lesson 5-7 Completing the Square

4 Problem 6 TEKS Process Standard (1)(D) Writing in Vertex Form What should be your first step? Complete the square. What is y = x 2 + 4x 6 in vertex form? Name the vertex, axis of symmetry, direction of opening, and y-intercept. y = x 2 + 4x - 6 y = x 2 + 4x Add ( 4 2 ) 2 = 2 2 to complete the square. Also, subtract 2 2 to leave the function unchanged. y = (x + 2) y = (x + 2) 2-10 Factor the perfect square trinomial. Simplify. The vertex is (-2, -10). The axis of symmetry is x = -2. The parabola opens upward. The y-intercept is (0, - 6). Check with a graphing calculator. Plot1 Plot2 Plot3 X \Y1 = X 2 +4X 6 4 \Y2 = 3 \Y3 = 2 1 \Y4 = 0 \Y5 = 1 \Y6 = Minimum 2 \Y7 = X= 2 Y= 10 Y1= 10 Y Problem 7 Analyzing a Parabola What are the vertex, focus, and directrix of the parabola with equation y = x 2 4x + 8? The equation of the parabola vertex focus directrix Find c, h, and k. Use these values to find the vertex, focus, and directrix. First, complete the square to get the equation in vertex form. How can you change the equation to an equivalent form? Subtract the same value outside the parentheses that you added inside the parentheses. y = x 2-4x + 8 Standard form y = ax 2 + bx + c y = (x 2-4x + 4) y = (x - 2) Note that, in this case, 1 4c = 1, so c = The vertex (h, k) is (2, 4). The focus (h, k + c) is (2, 4.25). The directrix y = k - c is y = Add ( 1 2 # -4) 2 inside parentheses; subtract it outside. Vertex form y = 1 4c (x - h)2 + k PearsonTEXAS.com 193

5 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Solve each equation by finding square roots. For additional support when completing your homework, go to PearsonTEXAS.com. 1. 5x 2 = x 2-4 = x 2 = x 2 = x 2-15 = x 2-40 = 0 7. Apply Mathematics (1)(A) A rectangular swimming pool is 6 ft deep. One side of the pool is 2.5 times longer than the other. The amount of water needed to fill the swimming pool is 2160 cubic feet. Find the dimensions of the pool. Solve each equation. 8. x 2 + 6x + 9 = 1 9. x 2-4x + 4 = x 2 + 8x + 16 = x x + 1 = x 2-30x = x x + 16 = 36 Justify Mathematical Arguments (1)(G) Complete the square. Justify your answer. 14. x x x 2 - x x 2-24x x x m 2-3m x 2 + 4x + Solve each quadratic equation by completing the square. 20. x 2 + 6x - 3 = x 2-12x + 7 = x 2-3x = x x 2 + 2x - 5 = x x 2-12x - 2 = x x = 12 Rewrite each equation in vertex form. Then find the vertex, focus, directrix, axis of symmetry, direction of opening, and y-intercept. 26. y = x 2 + 4x y = 2x 2-6x y = -x 2 + 4x The area of the rectangle shown is 80 square inches. What is the value of x? Find the value of k that would make the left side of each equation a perfect square trinomial. 2x x x 2 + kx + 25 = x 2 - kx = x 2 - kx = x 2 + kx = x 2 - kx + 4 = x 2 - kx + 49 = Use Representations to Communicate Mathematical Ideas (1)(E) The table shows some possible dimensions of rectangles with a perimeter of 100 units. Copy and complete the table. a. Plot the points (width, area). Find a model for the data set. b. What is another point in the data set? Use it to verify your model. c. What is a reasonable domain for this function? Explain. d. Find the maximum possible area. What dimensions yield this area? e. Find a function for area in terms of width without using the table. Do you get the same model as in part (a)? Explain. Width Length Area Lesson 5-7 Completing the Square

6 Solve each quadratic equation by completing the square. 37. x 2 + 5x - 3 = x 2 + 3x = x 2 - x = x 2 + x - 1 = x 2-4x = x 2 - x = x x = x x = x 2 + x = x 2 + 2x + 4 = x 2-6x = x 2-0.6x = Apply Mathematics (1)(A) The quadratic function h = -0.01x x + 2 models the height of a punted football. The horizontal distance in feet from the point of impact with the kicker s foot is x, and h is the height of the ball in feet. a. Write the function in vertex form. What is the maximum height of the punt? b. The nearest defensive player is 5 ft horizontally from the point of impact. How high must the player reach to block the punt? c. Suppose the ball was not blocked but continued on its path. How far down the field would the ball go before it hit the ground? Solve for x in terms of a x 2 - ax = 6a x 2 + ax = a a 2 x 2-8ax = a 2 x 2 + 8ax + 3 = x 2 + ax 2 = 9x + 9a 55. 6a 2 x 2-11ax = Solve x 2 = (6 12)x + 7 by completing the square. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Rewrite each equation in vertex form. Then find the vertex of the graph. 57. y = -4x 2-5x y = 1 2 x 2-5x y = x x TEXAS Test Practice 60. The graph of which inequality has its vertex at , -52? A. y 6 0 2x C. y 7 0 2x B. y 6 0 2x D. y 7 0 2x Which number is a solution of x 0 = 9 + x? F. -3 G. 0 H. 3 J Joanne tosses an apple seed on the ground. It travels along a parabola with the equation y = -x Assume the seed was thrown from a height of 4 ft. How many feet away from Joanne will the apple seed land? A. 1 ft B. 2 ft C. 4 ft D. 8 ft 63. List the steps for solving the equation x 2-9 = -8x by the method of completing the square. Explain each step. PearsonTEXAS.com 195