Math quadratic equations
3x(x-7)=0 Answer: x=0, x=7 ----------------------------------------- This equation is "true" if either 3x=0 or x-7 =0 Solutions? x=0, x=7
Solve the following: 3x(x-7)=0
x² + 5 x + 2 = 0 answer: x = (-5 + √17) / 2, x = (-5 - √17) / 2 ------------------------------------- a = 1 b = 5 c = 2 quadratic formula: x = (-b ± √(b²-4ac)) / 2a x = (-5 ± √(5²-(4*1*2))) / 2*1 x = (-5 ± √25-(4*1*2))) / 2*1 x = (-5 ± √25-8) / 2*1 x = (-5 ± √25-8) / 2 x = (-5 ± √17) / 2
solve this equation: x² + 5 x + 2 = 0
answer, length = 12 cm width = 5 cm --------------------------------------- width X length = area (x)(x+7)=60 x² + 7x = 60 x² + 7x - 60 = 0 quadratic equation x = (-b ± (√b²-4ac)) / 2a a = 1 b = 7 c = -60 x = (-7 ± √((7)²-(4*1*(-60))) / (2*1) x = (-7 ± √(49 + 240))/2 x = (-7 ± √289)/2 x = (-7 ± 17) / 2 x = 10/2, x = 24/2 x=5, x=12 check that works * yes: x+7 = 12, and 5 + 7 =12 * yes x * (x + 7) = 60; 5 * 12 = 60
Imagine a rectangle. The width is x (in cm) The length is x + 7 (in cm) the area is 60 cm² Find the dimensions
equation: x² -8x - 20 Answer: (x+2)(x-10)=0 solutions, x = -2, x = 10 -------------------------- x = (-b ± √(b²-4ac)) / 2a Remember--the entire front part of the formula goes into the numerator, to be divided by 2a. a = 1 b = -8 c = -20 (8 ± √((-8)²-(4*1*(-20))) / 2*1 = (8 ± √(64-(4*1*(-20))) / 2= (8 ± √(64-(-80)) / 2 = (8 ± √(64+80) / 2 = (8 ±√1(44) / 2 = (8 ± 12) / 2 = (8 + 12) / 2 = 10 (8 - 12) / 2 = -2
Solve x² -8x - 20 by quadratic formula.
equation: x² -8x - 20 = 0 answer: (x+2)(x-10)=0 x=10, x=-2 are solutions -------------------------------- a = 1 b = -8 c = -20 d = b/2a d = 8/(2*1) d = 8 e = c - (b²/4a) = e = -20 - ((-8²)/4) e = -20 - (64/4) e = -20 - 64/4 e = -20 - 16 e = -36 so 1(x+8)² - 36 = 0 (x+8)(x+8) = 36 x = -2 then look at the last term -2 * ? = 20 ? = 10
Solve by completing the square. x² -8x - 20 = 0
equation: x² -8x - 20 = 0 (x-10)(x+2)=0 x=10, x=-2 are solutions --------------------------- factor -20 some version of 1 and 20? no. wouldn't add to 8 some version of 2 and 10 yes! -10, +2 would add up to -8 (x-10)(x+2)=0 x=10, x=-2 are solutions
Solve by factoring. x² -8x - 20 = 0
equation: x² - 6x + 8 = 0 answer: (x-4) (x-2)=0 x=4, x=2 are solutions ------------------------ factor 8: would a combination of 8, 1 work? No, that would not add to 6. would a combination of 4,2 work? yes, -4 X -2 = 8 and -4 + -2 = -6
Solve by factoring: x² - 6x + 8 = 0
Problem: 3x² - 20x + 12 Answer: x=2/3, x=6 (x-2/3)(x-6)=0 ------------------------------------ use the quadratic formula to break down x = (-b ± √(b²-4ac)) / 2a Remember--the entire front part of the formula goes into the numerator, to be divided by 2a. a = 3 b = -20 c = 12 (-(-20) ± √(-20)²-(4*3*12)) / 2*3 ((20) ± √(400-144) / 6 (20 ± 16) / 6 x = 36/6 = 6 and x = 4/6 x = 6, x = 4/6
Solve the following: 3x² - 20x + 12
x² + 9x + 18 = 0 answer: (x+3)(x+6)=0 x= -3, x=-6 ----------------------- factor 1, 18? no 2, 9 no 3, 6 yes! 3x + 6x = 9x and 3*6=18 (x+3)(x+6)=0 x= -3, x=-6
Solve the problem: x² + 9x + 18 = 0
Problem: (4x + 10)(3x-7)=0 (or, 12x² + 2x -70) answer: x=2 1/3, x= - 2 1/2 --------------------------- The expression will be true if either of the terms in the parentheses is true. (4x + 10) = 0 OR (3x-7) = 0 4x + 10 = 0 4x = -10 x = -10/4 x = -5/2 or -2..5 3x - 7 = 0 3x = 7 x = 7/3 = 2 1/3
Solve the quadratic equation: (4x + 10)(3x-7)=0
equation: x² - 6x + 8 = 0 answer: (x-4) (x-2)=0 x=4, x=2 are solutions ------------------------------- x² - 6x = -8 (x-3) * (x-3) The squares would be (x-3) and (x-3) multiplied together they would be x² - 6x + 9 So, add 9 to each side x² - 6x + 9 = -8+9 = 1 (x-3)² = 1 x=4 x=2
solve by completing the square x² - 6x + 8 = 0
x² - 6x + 8 = 0 answer: (x-4) (x-2)=0 x=4, x=2 are solutions ---------------------------- x = (-b ± √(b²-4ac)) / 2a a=1 b=-6 c=8 so (6 ± √(6² - (4 * 1 * 8)) / 2*1 (6 ± √(36 - 32)) / 2 =(6 ± (√4) ) /2 = (6 ± 2) / 2 (6+2) / 2=4 (6-2) / 2 = 2 4, 2 check the original equation-- x-4 = 0 x-2 = 0 x=4, x=2
solve by using quadratic formula ---------------------------- x² - 6x + 8 = 0
25x² - 1 = 0 answer: x = 1/5, -1/5 (x + 1/5) (x-1/5) = 0 quadratic formula: x = (-b ± √(b²-4ac)) / 2a a = 25 b = 0 c = -1 x = (-b ± √(b²-4ac)) / 2a x = (0 ± √-(4*25*(-1)) / 2*25 x = (0 ± √(4*25)) / 2*25 x = (0 ± √100) / 2*25 x = (± √100) / 50 x = -10/50 x = 10/50 x = -1/5 x = + 1/5
solve the equation : 25x - 1 = 0
3x² + 8x - 3 answer (x-1/3)(x+3)=0 x = 1/3; x = -3 ------------------------------- x = (-b ± √(b²-4ac)) / 2a a = 3 b = 8 c = -3 x = (-8 ± √((-8)²-(4*3*(-3))) / (2*3) x = (-8 ± √(64-(4*3*(-3))) / (2*3) x = (-8 ± √(64-(-36)) / (2*3) x = (-8 ± √(64+36)) / (2*3) x = (-8 ± √100) / (2*3) x = (-8 ± √100)) / 6 x = (-8 ± 10)) / 6 x = 2/6 = 1/3; x = -3
solve the equation: 3x² + 8x - 3
