Whakaoti i te sqrt{6x-1}-sqrt{5x+4}=9

$$\sqrt { 6 x - 1 } - \sqrt { 5 x + 4 } = 9$$

Whakaoti mō x

$$x=18\sqrt{2459}+896\approx 1788.589491312$$

Ngā Upane Otinga

Me tango -\sqrt{5x+4} mai i ngā taha e rua o te whārite.

$$\sqrt{6x-1}=9+\sqrt{5x+4}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{6x-1}\right)^{2}=\left(9+\sqrt{5x+4}\right)^{2}$$

Tātaihia te \sqrt{6x-1} mā te pū o 2, kia riro ko 6x-1.

$$6x-1=\left(9+\sqrt{5x+4}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(9+\sqrt{5x+4}\right)^{2}.

$$6x-1=81+18\sqrt{5x+4}+\left(\sqrt{5x+4}\right)^{2}$$

Tātaihia te \sqrt{5x+4} mā te pū o 2, kia riro ko 5x+4.

$$6x-1=81+18\sqrt{5x+4}+5x+4$$

Tāpirihia te 81 ki te 4, ka 85.

$$6x-1=85+18\sqrt{5x+4}+5x$$

Me tango 85+5x mai i ngā taha e rua o te whārite.

$$6x-1-\left(85+5x\right)=18\sqrt{5x+4}$$

Hei kimi i te tauaro o 85+5x, kimihia te tauaro o ia taurangi.

$$6x-1-85-5x=18\sqrt{5x+4}$$

Tangohia te 85 i te -1, ka -86.

$$6x-86-5x=18\sqrt{5x+4}$$

Pahekotia te 6x me -5x, ka x.

$$x-86=18\sqrt{5x+4}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(x-86\right)^{2}=\left(18\sqrt{5x+4}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-86\right)^{2}.

$$x^{2}-172x+7396=\left(18\sqrt{5x+4}\right)^{2}$$

Whakarohaina te \left(18\sqrt{5x+4}\right)^{2}.

$$x^{2}-172x+7396=18^{2}\left(\sqrt{5x+4}\right)^{2}$$

Tātaihia te 18 mā te pū o 2, kia riro ko 324.

$$x^{2}-172x+7396=324\left(\sqrt{5x+4}\right)^{2}$$

Tātaihia te \sqrt{5x+4} mā te pū o 2, kia riro ko 5x+4.

$$x^{2}-172x+7396=324\left(5x+4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 324 ki te 5x+4.

$$x^{2}-172x+7396=1620x+1296$$

Tangohia te 1620x mai i ngā taha e rua.

$$x^{2}-172x+7396-1620x=1296$$

Pahekotia te -172x me -1620x, ka -1792x.

$$x^{2}-1792x+7396=1296$$

Tangohia te 1296 mai i ngā taha e rua.

$$x^{2}-1792x+7396-1296=0$$

Tangohia te 1296 i te 7396, ka 6100.

$$x^{2}-1792x+6100=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1792 mō b, me 6100 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1792\right)±\sqrt{\left(-1792\right)^{2}-4\times 6100}}{2}$$

Pūrua -1792.

$$x=\frac{-\left(-1792\right)±\sqrt{3211264-4\times 6100}}{2}$$

Whakareatia -4 ki te 6100.

$$x=\frac{-\left(-1792\right)±\sqrt{3211264-24400}}{2}$$

Tāpiri 3211264 ki te -24400.

$$x=\frac{-\left(-1792\right)±\sqrt{3186864}}{2}$$

Tuhia te pūtakerua o te 3186864.

$$x=\frac{-\left(-1792\right)±36\sqrt{2459}}{2}$$

Ko te tauaro o -1792 ko 1792.

$$x=\frac{1792±36\sqrt{2459}}{2}$$

Nā, me whakaoti te whārite x=\frac{1792±36\sqrt{2459}}{2} ina he tāpiri te ±. Tāpiri 1792 ki te 36\sqrt{2459}.

$$x=\frac{36\sqrt{2459}+1792}{2}$$

Whakawehe 1792+36\sqrt{2459} ki te 2.

$$x=18\sqrt{2459}+896$$

Nā, me whakaoti te whārite x=\frac{1792±36\sqrt{2459}}{2} ina he tango te ±. Tango 36\sqrt{2459} mai i 1792.

$$x=\frac{1792-36\sqrt{2459}}{2}$$

Whakawehe 1792-36\sqrt{2459} ki te 2.

$$x=896-18\sqrt{2459}$$

Kua oti te whārite te whakatau.

$$x=18\sqrt{2459}+896 x=896-18\sqrt{2459}$$

Whakakapia te 18\sqrt{2459}+896 mō te x i te whārite \sqrt{6x-1}-\sqrt{5x+4}=9.

$$\sqrt{6\left(18\sqrt{2459}+896\right)-1}-\sqrt{5\left(18\sqrt{2459}+896\right)+4}=9$$

Whakarūnātia. Ko te uara x=18\sqrt{2459}+896 kua ngata te whārite.

$$9=9$$

Whakakapia te 896-18\sqrt{2459} mō te x i te whārite \sqrt{6x-1}-\sqrt{5x+4}=9.

$$\sqrt{6\left(896-18\sqrt{2459}\right)-1}-\sqrt{5\left(896-18\sqrt{2459}\right)+4}=9$$

Whakarūnātia. Ko te uara x=896-18\sqrt{2459} kāore e ngata ana ki te whārite nā te mea e rerekē ngā tohu o te taha maui me te taha katau.

$$99-2\times 2459^{\frac{1}{2}}=9$$

Whakakapia te 18\sqrt{2459}+896 mō te x i te whārite \sqrt{6x-1}-\sqrt{5x+4}=9.

$$\sqrt{6\left(18\sqrt{2459}+896\right)-1}-\sqrt{5\left(18\sqrt{2459}+896\right)+4}=9$$

Whakarūnātia. Ko te uara x=18\sqrt{2459}+896 kua ngata te whārite.

$$9=9$$

Ko te whārite \sqrt{6x-1}=\sqrt{5x+4}+9 he rongoā ahurei.

$$x=18\sqrt{2459}+896$$