x²+6x+9=49 x²+6x-40=0 x1=-6/2 - Square root of ((6/2)²+40) x1=-3 - 7 x1= -10 x2=-6/2 + Square root of ((6/2)²+40) x2=-3 + 7 x2= 4
4-2 isn't 3, it's 2. 3+2 is 5.
if you mean X²+8X-5=0: X1=-8/2 - Square root of ((8/2)²+5) X1=-4 - Square root of 21 X1 is about -8.58 X2=-8/2 + Square root of ((8/2)²+5) X2=-4 + Square root of 21 X2 is about 0.58
Equals 4, Not 32 + 2 equals 4, not 3.No, 2 plus 2 is 4.
x-3y+2 = 4 -3y = -x+4-2 -3y = -x+2 y = 1/3x-2/3 Slope or gradient of the perpendicular line is -3 Use y-y1 = -3(x-x1) to find the perpendicular equation
x²+2x-15=0 x1=-2/2 - Square root of ((2/2)²+15) x1=-1-4 x1=-5 x2=-2/2 + Square root of ((2/2)²+15) x2=-1+4 x2=3
x²+6x+9=49 x²+6x-40=0 x1=-6/2 - Square root of ((6/2)²+40) x1=-3 - 7 x1= -10 x2=-6/2 + Square root of ((6/2)²+40) x2=-3 + 7 x2= 4
It is false-apex
√[(X2-X1)2+(Y2-Y1)2]minus a negative is plus a positive√[(5+3)2+(-2-4)2]plus a negative is minus a positive√(82-62)√(64-36)√28 or 2√7
4
3
Let (x1, y1) = (-3, -4) and (x2, y2) = (3, -2). Thus, the slope is 1/3. m = (y2 - y1)/(x2 - x1) = (-2 - -4)/(3 - -3) = (-2 + 4)/(3 + 3) = 2/6 = 1/3.
(3x4) / 2 - 1 [(4-2)x1]+3 [(4x2)/2]+1
y=x2+3 x1=1 x2=2 y(x1) = 1*1+3 = 4 y(x2) = 2*2+3 = 7 x2/x1 = 2, While y2/y1 = 7/4 !=2, and thus the function is nonlinear.
Intro/outro: e-|-7-8-------5-7------------------------| B-|-7-8-------3-5------5-7------8-5------| G-|--------------------2-4---------------| D-|-----------------------------0-2------| A-|--------------------------------------| E-|--------------------------------------| Riff 1: P.M. P.M.... e-|-------------------| B-|-------------------| G-|-------------------| D-|-------------------| A-|-/8-----/8---------| E-|-/6-0-0-/6-0-0-0-0-| Riff 2: P.M e-|-------------------| B-|-------------------| G-|-------------------| D-|-----------2-----5-| A-|-4---2---4-0-4---3-| E-|-2---0---2---2-2---| Add 1: e-|--------| B-|--------| G-|--------| D-|-4------| A-|-4------| E-|-2------| Add 2: e-|-------| B-|-------| G-|-------| D-|-5---4-| A-|-3---2-| E-|-------| Riff 3: P.M P.M....| e-|-------------------| B-|-------------------| G-|-------------------| D-|-/5-----/5---------| A-|-/3-----/3---------| E-|----2-2----2-2-2-2-| Chorus: P.M...........| P.M.........| e-|-----------------------------------------------------------| B-|-----------------------------------------------------------| G-|-----------------------------------------------------------| D-|-6---5-4---6-----------------6---5-4---6-------------------| A-|-4---3-2---4-0-0-0-0-0-0-0-0-4---3-2---4-0-0-0-0-0-0-0-2br-| E-|-----------------------------------------------------------| Riff 4: P.M |.| |.....| |-------| e-|-------------------------------------------------| B-|-------------------------------------------------| G-|-------------------------------------------------| D-|-9\8\7---5-----4---------------------------------| A-|-7\6\5---3-----2-5-----------5-2-----------5-2---| E-|-------0---0-0---3---0-0-0-0-3-0-0-0-0-0-0-3-0---| Riff 5: P.M...| |.........| |.........| |.........| e-|----------------------------------------------------------- B-|----------------------------------------------------------- G-|----------------------------------------------------------- D-|---------5---------------5---------------4---------------4- A-|-4-4-4-4-3---3-3-3-3-3-3-3---3-3-3-3-3-3-2---2-2-2-2-2-2-2- E-|----------------------------------------------------------- P.M.......| |.........| |.........| |.........| --------------------------------------------------------------- --------------------------------------------------------------- --------------------------------------------------------------- -------------3---------------3---------------2---------------2- -2-2-2-2-2-2-1---1-1-1-1-1-1-1---0-0-0-0-0-0-0---0-0-0-0-0-0-0- --------------------------------------------------------------- P.M....| |.........| |..........| |..........| -------------------------------------------------------------| -------------------------------------------------------------| -------------------------------------------------------------| -------------------------------------------------------------| ---------3---------------3---------------5---------------8---| -1-1-1-1-1---1-1-1-1-1-1-1---1-1-3-3-3-3-3---3-3-6-6-6-6-6---| Riff 6: P.M. e-|----------------| B-|----------------| G-|---------5------| D-|-9\8\7---5------| A-|-7\6\5---3------| E-|-------0--------| Add 3: e-|--------| B-|--------| G-|-4------| D-|-4------| A-|-2------| E-|--------| Riff 7: P.M |...| e-|----------------------------------| B-|----------------------------------| G-|-------------5--------------------| D-|-2-----------5--------------------| A-|-2-----------3---3-----5/---7-7---| E-|-0-------0-0-----3-3-3-3/---5-5---| Riff 8: P.M....| |......| |.| |.| |.| e-|----------------------------------------------------------| B-|----------------------------------------------------------| G-|----------------------------------------------------------| D-|-------------7--------------------------------------------| A-|-4-----------5-----------7/---7---7-------7-------7-------| E-|-2---2-2-2-2-----5-5-5-5-5/---5---5---5-5-5---5-5-5---5-5-| Riff 9: e-|----------------------------------------| B-|----------------------------------------| G-|----------------------------------------| D-|----------------------------------------| A-|-7---7---7------5---5------3------------| E-|-5---5---5------3---3------1------------| Riff 10: P.M...................................| e-|-------------------------------------------------------| B-|-------------------------------------------------------| G-|-------------------------------------------------------| D-|-------------------------------------------4---5---4---| A-|---------4-5-----------4-5-----------4-5---2---3---2---| E-|-0-0-0-0-2-3---0-0-0-0-2-3---0-0-0-0-2-3---------------| Riff 11: P.M....| |.........| |.........| |.........| e-|------------------------------------------------------------ B-|------------------------------------------------------------ G-|------------------------------------------------------------ D-|---------5---------------5---------------4---------------4-- A-|-3-3-3-3-3---3-3-3-3-3-3-3---3-3-2-2-2-2-2---2-2-2-2-2-2-2-- E-|------------------------------------------------------------ P.M........| |..........| |.........| |..| ----------------------------------------------------- ----------------------------------------------------- ----------------------------------------------------- -------------3---------------3---------------2------- -2-2-1-1-1-1-1---1-1-1-1-1-1-1---1-1-0-0-0-0-0---0-0- ----------------------------------------------------- P.M.....| |..| -----------------| -----------------| -----------------| ---------2-------| -0-0-0-0-0---0-0-| -----------------| Riff 12: P.M....| P.M........| P.M........| P.M.......| e-|-----------------------------------------------------------| B-|-----------------------------------------------------------| G-|-----------------------------------------------------------| D-|---------10--------------10--------------7---------------8-| A-|-8-8-8-8-8---8-8-8-8-8-8-8---8-8-5-5-5-5-5---5-5-6-6-6-6-6-| E-|-----------------------------------------------------------| Add 4: e-|-------------| B-|-------------| G-|-------------| D-|-5---4---3---| A-|-3---2---1---| E-|-------------| Riff 13: e-|------------------------| B-|------------------------| G-|------------------------| D-|-----------2-----5------| A-|-4---2---4-0-4---3------| E-|-2---0---2---2----------| Riff 14: e-|---------------------| B-|---------------------| G-|---------------------| D-|-----------2---------| A-|-4---2---4-0-4---2---| E-|-2---0---2---2---0---| Riff 15: P.M. |.| |.| |.| e-|-------------------------------------------------------| B-|-------------------------------------------------------| G-|-------------------------------------------------------| D-|-----------2-----5-------5-------5-------5-------5\4---| A-|-4---2---4-0-4---3---3-3-3---3-3-3---3-3-3---3-3-3\2---| E-|-2---0---2---2-----------------------------------------| Add 5: P.M.......... .| e-|-----------------| B-|-----------------| G-|-----------------| D-|-----------------| A-|-----------------| E-|-0-0-0-0-0-0-0-0-| That was all the riffs, lads! This will show you when to play them: Intro x4 Riff 1 x12 Riff 2 x4 Add 1 x1 Add 2 x1 Riff 1 x4 Add 2 x1 Riff 1 x4 Add 2 x1 Riff 3 x4 Chorus x1 Riff 3 x2 Chorus x1 Riff 2 x4 Riff 1 x4 Add 2 x1 Riff 1 x4 Add 2 x1 Riff 3 x4 Chorus x1 Riff 3 x2 Chorus x1 Riff 4 x4 Riff 5 x1 Riff 4 x3 Riff 6 x1 Add 3 x1 Riff 7 x4 Riff 8 x4 Riff 9 x1 Riff 10 x4 Riff 11 x2 Riff 12 x1 Riff 11 x1 Riff 12 x1 Riff 4 x4 Riff 5 x1 Riff 4 x3 Riff 6 x1 Riff 1 x12 Add 4 x1 Add 1 x1 Riff 1 x4 Add 1 x1 Riff 1 x4 Add1 x1 Riff 3 x4 Chorus x1 Riff 3 x2 Chorus x1 Riff 2 x4 Riff 13 x1 Riff 14 x1 Riff 13 x1 Riff 14 x1 Riff 15 x1 Riff 14 x1 Add 5 x1 Riff 15 x1 Riff 14 x1 Add 5 x1 Outro x1
43
Let P(x1, y1), Q(x2, y2), and M(x3, y3).If M is the midpoint of PQ, then,(x3, y3) = [(x1 + x2)/2, (y1 + y2)/2]We need to verify that,√[[(x1 + x2)/2 - x1]^2 + [(y1 + y2)/2 - y1]^2] = √[[x2 - (x1 + x2)/2]^2 + [y2 - (y1 + y2)/2]^2]]Let's work separately in both sides. Left side:√[[(x1 + x2)/2 - x1]^2 + [(y1 + y2)/2 - y1]^2]= √[[(x1/2 + x2/2)]^2 - (2)(x1)[(x1/2 + x2/2)) + x1^2] + [(y1/2 + y2/2)]^2 - (2)(y1)[(y1/2 + y2/2)] + y1^2]]= √[[(x1)^2]/4 + [(x1)(x2)]/2 + [(x2)^2]/4 - (x1)^2 - (x1)(x2) + (x1)^2 +[(y1)^2]/4 + [(y1)(y2)]/2 + [(y2)^2]/4 - (y1)^2 - (y1)(y2) + (y1)^2]]= √[[(x1)^2]/4 - [(x1)(x2)]/2 + [(x2)^2]/4 + [(y1)^2]/4 - [(y1)(y2)]/2 + [(y2)^2]/4]]Right side:√[[x2 - (x1 + x2)/2]^2 + [y2 - (y1 + y2)/2]^2]]= √[[(x2)^2 - (2)(x2)[(x1/2 + x2/2)] + [(x1/2 + x2/2)]^2 + [(y2)^2 - (2)(y2)[(y1/2 + y2/2)] + [(y1/2 + y2/2)]^2]]= √[[(x2)^2 - (x1)(x2) - (x2)^2 + [(x1)^2]/4 + [(x1)(x2)]/2 + [(x2)^2]/4 + (y2)^2 - (y1)[(y2) - (y2)^2 + [(y1)^2]/4) + [(y1)(y2)]/2 + [(y2)^2]/4]]= √[[(x1)^2]/4 - [(x1)(x2)]/2 + [(x2)^2]/4 + [(y1)^2]/4 - [(y1)(y2)]/2 + [(y2)^2]/4]]Since the left and right sides are equals, the identity is true. Thus, the length of PM equals the length of MQ. As the result, M is the midpoint of PQ