Quadratic Review Answers

Math III Quadratics Review Standard form f(x) = ax2 + bx + c c Is the y-intercept of a quadratic, positive a (faces up like a U), negative o (faces down) Vertex Form f(x)=a(xh)²+k vertex(hk) Solutions (known as x-intercepts, zeroes, or roots) of a quadratic can be found four ways: Graphing - Graph the function in y=, 2", trace, zero 3 y(x-2)(x-1) 2 Solution 0x + 1)(x+1) 2 1 No real solutions -2-1 3 1234 1 2 3 4 Two Solutions One solution Imaginary Solutions Factoring-transform a quadratic from standard form into factored Ex/ Solve f(x) = 3x²+7x-6 Factor/(x)= (3x-2)(x+3) Set each factor equal to zero and solve for x 3x-2-0 3x=2 3 x+3=0 x=-3 Quadratic Formulax= 24 Ex/(x)=3x²+7x-6 x= use the same a, b, c from standard form a=3,b=7, and c=-6 -7土(7)²-4(3) (-6) 2(3) -7±11 6 2 So x== and x = -3 3 Completing the Square "If a 1, divide it from the equation Ex/x²-6x+25=0 x²-6x =-25 x²-6x+9-25+9 (x-3)2-16 x-3=±√-16 x-3=+41 x=3241 Ex/3x²-6x-90 3x²-6x9 x²-2x=3 x²-2x+1=3+1 (x-1)2=4 x-1=±2 x=3.-1 move the c value to the other side add() to buth sides 2 factor the left side take the square root to both sides simplify the right side solve for x Factor by grouping EX/8xy-2x+20y-5 2x(4y-1)5(41) split expression in half so a GCF can be pulled out and find the GCF of each side (4y-1)(2x+5) Pull out the common factor, the GCF's become a factor together Imaginary Numbers 1 is an imaginary number and occurs when a negative is under a square root 12-1 Examples 1. Simplify -8 212 2. Simplify -32 452 3. Simplify (-1+5)+(-2-31) 3+2 4. Simplify (3+28)(2-51) (6-1544-10(-1) 6--11i+10-16-4 5. What are the values of x and y when (5+41)(x+y)=(-1-31)? x=6, y=7 С. хиб, у Β. x-4, γέ D. x = 4, y = 7 -(x+y)=-1-3-5-40 -() (-7) 6. Which is equivalent to (4-31)+(6+1)? A 302 B. 50 42-121 D. 62-121 x+yi = 6+7 16-12-122+9i (7-2) +35+120=42-120 7. For which equation is the x-coordinate of the vertex at 47 f(x) = x²-8x+15 C. f(x) = x²+6x+8 B. f(x)=x²-4x+12 D. f(x)=x²-2x+2 b 20 8 2 8. Adrian is using 120 feet of fencing to enclose a rectangular area for her puppy. One side of the enclosure will be her house. The function f(x)=x(120-2x) represents the area of the enclosure. What is the greatest area that Adrian can enclase for fencing? A. 1650 ft 8.1800 ft C. 1980 ft D. 2140 ft -2x²+120x 18. A square and a rectangle have the same area. The length of the rectangle is five inches more than twice the length of the side of the square. The width of the rectangle is 6 inches less than the side of the square. Find the length of the side of the square. x²= (x-16) (2x+5) x²-2x²+5x-12x-30 2. X=10 2x+5 x²-7x-30 ( 2 x+3Xx-PO) 19. A cardboard box company has been contracted to manufacture open-top rectangular storage boxes for a manufacturing company. The company has 30 cm X 16 cm cardboard sheets. They plan to cut a square from each comer of the sheet and bend up the sides to form the box. If the company wants to make boxes with the largest possible volume: 1-30-2.x out(rex) Find the dimensions of the square being cut out. X=3.33 30-2x L=23.34 What are the dimensions of the box? What is the maximum volume of the box? (116-2x130-2x)) W=16-2.x 9.34 h=3.33 max 726 20. A box with an open top will be constructed from a rectangular piece of cardboard. The piece of cardboard is 8 inches by wide and 12 Inches long. The box will be constructed by cutting out equal squares of sides x at each corner and then folded up at the sides. 12 8 What is the entire domain for the function V(x) that gives the volume of the box as a function of x? X X(12-2xY8-2x 21. Abey Numkena is an interior designer. She has been asked to locate an oriental rug for a new corporate office. As a rule, the rug should cover of the total floor area with a uniform width 2 surrounding the rug. 42(192)=96 A= 192 if the dimensions of the room are 12 feet by 16 feet, write an equation to model the situation. (12-2x) (16-2x)-96 4x²-B6x+96 FAC Sketch a graph of the function. What are the dimensions of the rug? 12-2(2)=8 1112 = (2)2-م 1611 12- 21 8x12