Whakaoti i te sqrt{2x+7}-3x+1=x-1

$$\sqrt { 2 x + 7 } - 3 x + 1 = x - 1$$

Whakaoti mō x

$$x = \frac{\sqrt{129} + 9}{16} \approx 1.272363543$$

Ngā Upane Otinga

Me tango -3x+1 mai i ngā taha e rua o te whārite.

$$\sqrt{2x+7}=x-1-\left(-3x+1\right)$$

Hei kimi i te tauaro o -3x+1, kimihia te tauaro o ia taurangi.

$$\sqrt{2x+7}=x-1-\left(-3x\right)-1$$

Ko te tauaro o -3x ko 3x.

$$\sqrt{2x+7}=x-1+3x-1$$

Pahekotia te x me 3x, ka 4x.

$$\sqrt{2x+7}=4x-1-1$$

Tangohia te 1 i te -1, ka -2.

$$\sqrt{2x+7}=4x-2$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{2x+7}\right)^{2}=\left(4x-2\right)^{2}$$

Tātaihia te \sqrt{2x+7} mā te pū o 2, kia riro ko 2x+7.

$$2x+7=\left(4x-2\right)^{2}$$

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

$$2x+7=16x^{2}-16x+4$$

Tangohia te 16x^{2} mai i ngā taha e rua.

$$2x+7-16x^{2}=-16x+4$$

Me tāpiri te 16x ki ngā taha e rua.

$$2x+7-16x^{2}+16x=4$$

Pahekotia te 2x me 16x, ka 18x.

$$18x+7-16x^{2}=4$$

Tangohia te 4 mai i ngā taha e rua.

$$18x+7-16x^{2}-4=0$$

Tangohia te 4 i te 7, ka 3.

$$18x+3-16x^{2}=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-16x^{2}+18x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -16 mō a, 18 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-16\right)\times 3}}{2\left(-16\right)}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\left(-16\right)\times 3}}{2\left(-16\right)}$$

Whakareatia -4 ki te -16.

$$x=\frac{-18±\sqrt{324+64\times 3}}{2\left(-16\right)}$$

Whakareatia 64 ki te 3.

$$x=\frac{-18±\sqrt{324+192}}{2\left(-16\right)}$$

Tāpiri 324 ki te 192.

$$x=\frac{-18±\sqrt{516}}{2\left(-16\right)}$$

Tuhia te pūtakerua o te 516.

$$x=\frac{-18±2\sqrt{129}}{2\left(-16\right)}$$

Whakareatia 2 ki te -16.

$$x=\frac{-18±2\sqrt{129}}{-32}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{129}}{-32} ina he tāpiri te ±. Tāpiri -18 ki te 2\sqrt{129}.

$$x=\frac{2\sqrt{129}-18}{-32}$$

Whakawehe -18+2\sqrt{129} ki te -32.

$$x=\frac{9-\sqrt{129}}{16}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{129}}{-32} ina he tango te ±. Tango 2\sqrt{129} mai i -18.

$$x=\frac{-2\sqrt{129}-18}{-32}$$

Whakawehe -18-2\sqrt{129} ki te -32.

$$x=\frac{\sqrt{129}+9}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{9-\sqrt{129}}{16} x=\frac{\sqrt{129}+9}{16}$$

Whakakapia te \frac{9-\sqrt{129}}{16} mō te x i te whārite \sqrt{2x+7}-3x+1=x-1.

$$\sqrt{2\times \frac{9-\sqrt{129}}{16}+7}-3\times \frac{9-\sqrt{129}}{16}+1=\frac{9-\sqrt{129}}{16}-1$$

Whakarūnātia. Ko te uara x=\frac{9-\sqrt{129}}{16} kāore e ngata ana ki te whārite nā te mea e rerekē ngā tohu o te taha maui me te taha katau.

$$-\frac{15}{16}+\frac{7}{16}\times 129^{\frac{1}{2}}=-\frac{7}{16}-\frac{1}{16}\times 129^{\frac{1}{2}}$$

Whakakapia te \frac{\sqrt{129}+9}{16} mō te x i te whārite \sqrt{2x+7}-3x+1=x-1.

$$\sqrt{2\times \frac{\sqrt{129}+9}{16}+7}-3\times \frac{\sqrt{129}+9}{16}+1=\frac{\sqrt{129}+9}{16}-1$$

Whakarūnātia. Ko te uara x=\frac{\sqrt{129}+9}{16} kua ngata te whārite.

$$-\frac{7}{16}+\frac{1}{16}\times 129^{\frac{1}{2}}=\frac{1}{16}\times 129^{\frac{1}{2}}-\frac{7}{16}$$

Ko te whārite \sqrt{2x+7}=4x-2 he rongoā ahurei.

$$x=\frac{\sqrt{129}+9}{16}$$