Whakaoti i te 3x+54/6x^{2+3x}+4x^2+9/4x^2-1=x+3/2/3x-frac{8}{3}*frac{x^2}{(1-4x^2)}

$$\frac { 3 x + 54 } { 6 x ^ { 2 } + 3 x } + \frac { 4 x ^ { 2 } + 9 } { 4 x ^ { 2 } - 1 } = \frac { x + \frac { 3 } { 2 } } { 3 x } - \frac { 8 } { 3 } \cdot \frac { x ^ { 2 } } { ( 1 - 4 x ^ { 2 } ) }$$

Whakaoti mō x

$$x=\frac{15}{38}\approx 0.394736842$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -\frac{1}{2},0,\frac{1}{2} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3x\left(2x-1\right)\left(2x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 6x^{2}+3x,4x^{2}-1,3x,3,1-4x^{2}.

$$\left(2x-1\right)\left(3x+54\right)+3x\left(4x^{2}+9\right)=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)xx^{2}$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-1 ki te 3x+54 ka whakakotahi i ngā kupu rite.

$$6x^{2}+105x-54+3x\left(4x^{2}+9\right)=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)xx^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te 4x^{2}+9.

$$6x^{2}+105x-54+12x^{3}+27x=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)xx^{2}$$

Pahekotia te 105x me 27x, ka 132x.

$$6x^{2}+132x-54+12x^{3}=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)xx^{2}$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 1 me te 2 kia riro ai te 3.

$$6x^{2}+132x-54+12x^{3}=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)x^{3}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4x^{2}-1 ki te x+\frac{3}{2}.

$$6x^{2}+132x-54+12x^{3}=4x^{3}+6x^{2}-x-\frac{3}{2}-\frac{8}{3}\left(-3\right)x^{3}$$

Whakareatia te \frac{8}{3} ki te -3, ka -8.

$$6x^{2}+132x-54+12x^{3}=4x^{3}+6x^{2}-x-\frac{3}{2}-\left(-8x^{3}\right)$$

Ko te tauaro o -8x^{3} ko 8x^{3}.

$$6x^{2}+132x-54+12x^{3}=4x^{3}+6x^{2}-x-\frac{3}{2}+8x^{3}$$

Pahekotia te 4x^{3} me 8x^{3}, ka 12x^{3}.

$$6x^{2}+132x-54+12x^{3}=12x^{3}+6x^{2}-x-\frac{3}{2}$$

Tangohia te 12x^{3} mai i ngā taha e rua.

$$6x^{2}+132x-54+12x^{3}-12x^{3}=6x^{2}-x-\frac{3}{2}$$

Pahekotia te 12x^{3} me -12x^{3}, ka 0.

$$6x^{2}+132x-54=6x^{2}-x-\frac{3}{2}$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$6x^{2}+132x-54-6x^{2}=-x-\frac{3}{2}$$

Pahekotia te 6x^{2} me -6x^{2}, ka 0.

$$132x-54=-x-\frac{3}{2}$$

Me tāpiri te x ki ngā taha e rua.

$$132x-54+x=-\frac{3}{2}$$

Pahekotia te 132x me x, ka 133x.

$$133x-54=-\frac{3}{2}$$

Me tāpiri te 54 ki ngā taha e rua.

$$133x=-\frac{3}{2}+54$$

Tāpirihia te -\frac{3}{2} ki te 54, ka \frac{105}{2}.

$$133x=\frac{105}{2}$$

Whakawehea ngā taha e rua ki te 133.

$$x=\frac{\frac{105}{2}}{133}$$

Tuhia te \frac{\frac{105}{2}}{133} hei hautanga kotahi.

$$x=\frac{105}{2\times 133}$$

Whakareatia te 2 ki te 133, ka 266.

$$x=\frac{105}{266}$$

Whakahekea te hautanga \frac{105}{266} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 7.

$$x=\frac{15}{38}$$