UNIT 6 QUADRATIC EQUATIONS Date Lesson TOPIC Homework Apr. 4 Apr. 6 6.1 6.1 6. 6.3 Solving Quadratic Equations Pg. 319 # 1,, (4 8)ce, 10, 11, 14, 16b Completing the Square Pg. 331 # 1, 5 8, 10, 11, 13, 16 Apr. 7 Apr. 8 6.3 6.4 6.4 6.5 May 1 6.5 6.6 May 6.6 6.6 The Quadratic Formula CTS QUIZ The Nature of the Roots of a Quadratic Equation CTS QUIZ Solving Problems using Quadratic Models CTS QUIZ Solving Problems using Quadratic Models Quiz (6.1 6.4) Pg. 343 #, 4 6, 9, 1-14 Pg. 350 #, (3 5)ace, 6 10 Pg. 357 # 7, 11 WS 6.6 May 3 6.7 Review for Unit 6 Test Pg. 361 # (1, 3, 4, 8)ace, 5ab, 6, 7, 9, 10, (11, 1)ace, 13-17 May 5 6.8 TEST- UNIT 6
MPM D Lesson 6.1 Solving Quadratic Equations 5 4 3 1 y The solutions to a quadratic equation are the values of x at which the graph of the quadratic cross the axis of the independent (x) variable [ie: where the dependent variable(y) = 0]. 5 4 3 1 1 3 4 5 x 1 3 4 5 Every quadratic equation has solutions, however, those two solutions may be the same value or one or both of them may be imaginary numbers (ie: not real numbers.) The solutions of a quadratic equation are also known as ROOTS or ZEROS. ALGORITHM Set the quadratic = 0. Factor the quadratic completely. Set each factor = 0. Solve for the variable in each linear equation Check your answer(s) in the original equation. Ex. Solve each of the following. a) ( x 1)( x ) 0 b) x ( x 3) 0 x(x 3) = 0 c) y 3y d) 3( x x) 1 x
e) t 11t 5 f) 10x 16x 6 g) 5n 8n 0 h) 5x 40x 16 0 4 i) x x 1 0 j) x 5 x 8 4 Pg. 319 # 1,, (4 8)ce, 10, 11, 14, 16b
MPM D Lesson 6. Completing the Square Until now if we wanted to change the equation of a quadratic in standard form form y ax bx c into vertex y a( x h) k we would use factoring or partial factoring to find the axis of symmetry and then use that to find the y-value of the vertex. A method that many find to be quicker is called COMPLETING THE SQUARE. YOU ARE RESPONSIBLE FOR THIS METHOD AND SHOULD EXPECT IT TO BE ON FUTURE EVALUATIONS. Ex. 1 Change the following into vertex form by completing the square. a) y 5x 0 x b) y x 8x 15
Ex. Rewrite each of the following in vertex form and state the vertex. a) y x 1x 7 b) y x x 3 c) 1 y x 4 x 1 Pg. 331 # 1, 5 8, 10, 11, 13, 16
MPM D Lesson 6.3 The Quadratic Formula Not all quadratic equations can be solved by factoring. When dealing with these equations, we can Complete the square and then rearrange to solve for x. This can be time consuming. ie: Solve x x 5 0 by completing the square THERE HAS TO BE A FASTER WAY!
THERE IS! We can determine a general formula by completing the square on the standard equation of a quadratic and rearrange for x. This will give us a formula to solve for x for any quadratic equation in standard form ax + bx + c = 0. DERIVATION OF THE QUADRATIC FORMULA
ie: Solve x x 5 0 using the quadratic formula Ex. 1 Solve for x in each of the following. When necessary, round your solutions correct to decimal places. a) x 10 8 b) 3x (5x 4) x x 4( x 3)
Ex. The length of a photograph is 1 cm more than the width. The area of the photo is 45 cm. Determine the dimensions of the photograph, correct to two decimal places. Pg. 343 #, 4 6, 9, 1-14
Pg. 335 # 1 4, 6-9
MPM D Lesson 6.4 The Nature of the Roots of a Quadratic Equation Relation & Vertex a) y = x + x 9 Roots Use Quadratic Formula Number of Real Roots Sketch y 10 Value of b 4ac 8 6 4 5 4 3 1 1 3 x 4 6 8 10 Vertex = (-1, -10) b) y = x + 6x + 9 10 8 6 4 y 6 5 4 3 1 1 x 4 6 8 10 Vertex = (-3, 0) c) y = x + 4x + 7 14 1 10 8 6 4 y 6 5 4 3 1 1 x 4 6 Vertex = (-, 3) Results a) b 4ac is called the. b) If b 4ac > 0, then the quadratic has. a) If b 4ac = 0, then the quadratic has. If b 4ac < 0, then the quadratic has. 9or 15... are called numbers.
Determine the number of real roots for each of the quadratics from the table on page 1. Number of Real DISCRIMINANT ( b 4ac ) Roots (zeros/solutions) a) b) c) Ex. 1 Use the discriminant to determine the number of solutions of: a) 9x 4 x 49 0 b) 3x 5x 10
Ex. For what value of k will kx 5x 6 0 have no zeros? Ex. An arrow is released with an initial speed of 39. m/s. It travels according to t 4.9t 39.t 1. 3 h, where h is the height reached, in metres, and t is the time taken, in seconds. Will the arrow ever reach a height of 80 metres? Pg. 350 #, (3 5)ace, 6 10
MPM D Lesson 6.5 Solving Problems using Quadratic Models Strategy Read the question carefully, making note of what information is given and what you must find. Introduce variables to represent any unknowns. Use information in the question to set up a quadratic equation. Solve the equation. (i) set = 0 (ii) factor or use the quadratic formula Find the answer to the problem. Be sure to check your answer Write a closing statement. Ex. 1 The population of a city is modeled by the relation P 0.5t 10t 00, where P is the population in thousands and t is in years after the year 000. a) What was the population in 000? b) What was the population in 1994? c) When will the population reach 46 500?
Ex. The length of a rectangular backyard pool is 7 m more than twice the width. If the area is 10 m, find the dimensions of the pool, correct to 1 decimal place.
Ex. 3 A rectangular pool measuring 10 m x 5 m is to have a deck of uniform width built around it. If the area of the pool is equal to the area of the deck, how wide is the deck, correct to 1 dec. place?
Ex. 4 The volunteers at a food bank are arranging a concert to raise money. They have to pay a set fee to the musicians, plus an additional fee to the concert hall for each person attending the concert. The relation P = n + 580n 48 000 models the profit, P, in dollars, for the concert, where n is the number of tickets sold. a) Calculate the number of tickets they must sell to break even. b) Determine the number of tickets they must sell to maximize the profit. Pg. 357 # 7, 11
MPM D Lesson 6.6 Solving Problems using Quadratic Models II Ex. 1 Two numbers have a sum of 10. If the numbers are squared and added together the result is 58. Find the numbers.
Ex. Alexandre was practising his 10 m platform dive. Because of gravity, the relation between his height, h, in metres, and the time, t, in seconds, after he dives is quadratic. If Alexandre reached a maximum height of 11.5 m after 0.5 s, how long was he above the water after he dove? WS 6.6