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x³+x=⅝ --> x(x²+1)=⅛(4+1) =½×¼(4+1) =½(1+¼ ) =½[1+(½)²]Thus x=½ 0:13
7:30 you missed the sign of 1/27
No, he didnt
My motto is no pain, no pain 😅
Simplification of x in the 2nd method is a problem in its own right.
👍8 x^3+8 x - 5 = 0there is only one sign change in f(x) and no sign change in f(-x) hence by Descartes' rule of signs equation has only one real root and it is positive.by RRT we obtain root , x = 1/2
problemx³ + x = 5/8Cubic formulax = a + ba³ + b³ = 5/8-3ab = 1ab = -1/3a³ b ³ = -1/27b³ = -1/(27 a³)a³ -1/(27 a³) = 5/827 (a³)²-(5/8)27 (a³)-1 = 0a³ = { (5/8)27 ± √[(5/8)² 27² +4(27)] }/54Δ = (5/8)² 27² +4(27) = (27)(25•27/64 +256/64) = 9•3/(64) (25•27+256) = 9•3/(64) 931a³ = { (5/8)27 ± (3/8)√[3•931] }/54 = { 135 ± 21√57} / 432 = { 45 ± 7√57} / 144x = a + b= ∛ [ (45 + 7√57) / 144 ] + ∛ [ (45 - 7√57) / 144 ] = 1/2This is one real root.x -1/2 a factor. x³ + x - 5/8 = 0(x -1/2) x² +(x-1/2) x/2+(5/4)(x-1/2) = 0(x-1/2)(x² +x/2 +5/4) = 0ZPP and Quadraticx = (-1/2 ± √[(1/4)-5]}/2 = -1/4 ± 1/4 i √19 = (-1 ± i √19 )/4answerx ∈ { 1/2, (-1 - i √19 )/4, (-1 + i √19 )/4 }
😂
x³+x=⅝ --> x(x²+1)=⅛(4+1)
=½×¼(4+1)
=½(1+¼ )
=½[1+(½)²]
Thus x=½ 0:13
7:30 you missed the sign of 1/27
No, he didnt
My motto is no pain, no pain 😅
Simplification of x in the 2nd method is a problem in its own right.
👍
8 x^3+8 x - 5 = 0
there is only one sign change in f(x) and no sign change in f(-x) hence by Descartes' rule of signs equation has only one real root and it is positive.
by RRT we obtain root , x = 1/2
problem
x³ + x = 5/8
Cubic formula
x = a + b
a³ + b³ = 5/8
-3ab = 1
ab = -1/3
a³ b ³ = -1/27
b³ = -1/(27 a³)
a³ -1/(27 a³) = 5/8
27 (a³)²-(5/8)27 (a³)-1 = 0
a³ = { (5/8)27 ± √[(5/8)² 27² +4(27)] }/54
Δ = (5/8)² 27² +4(27)
= (27)(25•27/64 +256/64)
= 9•3/(64) (25•27+256)
= 9•3/(64) 931
a³ = { (5/8)27 ± (3/8)√[3•931] }/54
= { 135 ± 21√57} / 432
= { 45 ± 7√57} / 144
x = a + b
= ∛ [ (45 + 7√57) / 144 ] + ∛ [ (45 - 7√57) / 144 ]
= 1/2
This is one real root.
x -1/2 a factor.
x³ + x - 5/8 = 0
(x -1/2) x² +(x-1/2) x/2+(5/4)(x-1/2) = 0
(x-1/2)(x² +x/2 +5/4) = 0
ZPP and Quadratic
x = (-1/2 ± √[(1/4)-5]}/2
= -1/4 ± 1/4 i √19
= (-1 ± i √19 )/4
answer
x ∈ { 1/2,
(-1 - i √19 )/4,
(-1 + i √19 )/4 }
😂