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unit 12 foldable for class ink.notebook May 08, 2017 Solving Quadratic Equations Solving Quadratic Equations 3x2 – 8 = 22 +8 +8 2 3x = 30 3 3 x2 = 10 x =+ - 10 1) get x2 by itself Add or subtract constant to each side 5(x + 1)2 = 80 5 5 1) get (x + 1)2 by itself (x + 1)2 = 16 2) Take the square root of each side +4 x + 1 =-1 -1 +4 x = -1 - multiply or divide coefficient 2) Take the square root of each side - 1 + 4 x = 3 1 - 4 x = -5 - multiply or divide coefficient 3) get x by itself Add or subtract constant to each side 4) get 2 answers if possible Completing the Square x + 6x – 16 = 0 + 16 + 16 1) move the constant to the x2 + 6x + ___ = 16 + ___ other side & add blanks 2 x + 6x + 9 = 16 + 9 6 2) Divide by 2 and square 2 it to fill in blanks 2 3 2 (x + 3) = 25 2 Solve each equation. Recall that those with an x2 will have two solutions. A) 3x2 – 18 = 30 B) 5(x – 6)2 = 250 This is your squared binomial (x + 3)2 = 25 x + 3 = ±5 -3 -3 x = -3 ± 5 -3 - 5 -3 + 5 x = 2 and x = –8 3) Take the square root of each side 4) add or subtract the constant to the other side and solve for x 1 unit 12 foldable for class ink.notebook Solve by completing the square. Show all work. Give the exact answer only. C) x2 – 8x = 20 Solve by completing the square. Show all work. Give the exact answer only. D) x2 + 6x + 4 = 10 Completing the Square - VERTEX FORM f(x) = x2 – 10x + 5 – 5 – 5 1) move the constant to the –5 + ___ = x2 – 10x + ___ other side & add blanks 2 – 5 + 25 = x – 10x + 25 –10 2) Divide by 2 and square it to fill in blanks 2 (–5)2 20 = (x – 5)2 -20 -20 3) add or subtract the 2 constant to the other f(x) = (x – 5) – 20 Vertex: (5, –20) May 08, 2017 Write each quadratic function in vertex form. Give the coordinates of the vertex and the axis of symmetry. Then tell whether the vertex is a maximum or a minimum. E) f(x) = x2 + 4x – 12 side 4) Put f(x) back it Axis of symmetry: x = 5 2 unit 12 foldable for class ink.notebook Write each quadratic function in vertex form. Give the coordinates of the vertex and the axis of symmetry. Then tell whether the vertex is a maximum or a minimum. May 08, 2017 G) Create a quadratic equation that has solutions of –5 and 8. Leave in factored form. F) f(x) = x2 – 14x + 24 H) A package of supplies is dropped from a helicopter hovering 200 meters above the ground. The attached parachute fails to open. The equation h = –4.9t2 + 200 models this situation. After how many seconds will the package reach the ground? Round to the nearest hundredth. The Discriminant ax2 + bx + c = 0 (must = 0) the discriminant = (b)2 - 4ac if (b)2 - 4ac > 0 (positive) then 2 distinct real solutions if (b)2 - 4ac = 0 then 1 real solution or a double root if (b)2 - 4ac < 0 (negative) then 0 real solutions, 2 complex solutions (imaginary) 3 unit 12 foldable for class ink.notebook Find the discriminant and tell the number AND type of solutions: I) 2x2 + 3x + 4 = 0 J) 9x 2 – 18x = –9 May 08, 2017 Find the discriminant and tell the number AND type of solutions: K) 2x2 + 5x = 4 The Quadratic Formula ax2 + bx + c = 0 (must = 0) L) An athlete throws a shot put and the height can be modeled by the equation h = –16t2 + 29t + 6. Determine if it ever reaches a height of 20 ft. x = – b ± (b)2 - 4ac 2a 4 unit 12 foldable for class ink.notebook May 08, 2017 Solve the following by using the Quadratic formula. The Quadratic Formula N) M) 2x2 + 5x – 7 = 0 n2 + 4n = –1 2x2 – 11x = 21 2x2 – 11x – 21 = 0 (must = 0) x = –(–11) ± x = x = (–11)2 - 4(2)(–21) 2(2) 11 ± 289 4 11 + 17 4 x = 7 = 11 ± 17 4 x = 11 – 17 4 x = –2 O) Julian kicked a soccer ball into the air with an initial upward velocity of 40 feet per second. The height h in feet of the ball above the ground can be modeled by h = –16t2 + 40t + 1, where t is the time in seconds after Julian kicked the ball. What is the height of the ball after 1 second? Find the time it takes the ball to reach the ground. Round to nearest hundredth. Dropped in feet: H = landing height in feet Launched or thrown in feet: – 16t2 + h0 time in seconds H= – initial height in feet 16t2 + v0t + h0 initial velocity in feet/sec 5 unit 12 foldable for class ink.notebook May 08, 2017 Q) The length of a rectangle is 9 more than the width. Write an expression for the area of the rectangle. P) A tennis ball is dropped from a height of 213 feet. How long does it take for the ball to hit the ground? The area is 90 square centimeters. Find the dimensions of it. 6