Whakaoti i te (17x-1)^2/13^2=(x-1)(2x)

$$\frac { ( 17 x - 1 ) ^ { 2 } } { 13 ^ { 2 } } = ( x - 1 ) ( 2 x )$$

Whakaoti mō x

$$x = \frac{13 \sqrt{137} + 152}{49} \approx 6.207369364$$

$$x=\frac{152-13\sqrt{137}}{49}\approx -0.003287731$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(17x-1\right)^{2}.

$$\frac{289x^{2}-34x+1}{13^{2}}=\left(x-1\right)\times 2x$$

Tātaihia te 13 mā te pū o 2, kia riro ko 169.

$$\frac{289x^{2}-34x+1}{169}=\left(x-1\right)\times 2x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te 2.

$$\frac{289x^{2}-34x+1}{169}=\left(2x-2\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-2 ki te x.

$$\frac{289x^{2}-34x+1}{169}=2x^{2}-2x$$

Whakawehea ia wā o 289x^{2}-34x+1 ki te 169, kia riro ko \frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}.

$$\frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}=2x^{2}-2x$$

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

$$\frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}-2x^{2}=-2x$$

Pahekotia te \frac{289}{169}x^{2} me -2x^{2}, ka -\frac{49}{169}x^{2}.

$$-\frac{49}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}=-2x$$

Me tāpiri te 2x ki ngā taha e rua.

$$-\frac{49}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}+2x=0$$

Pahekotia te -\frac{34}{169}x me 2x, ka \frac{304}{169}x.

$$-\frac{49}{169}x^{2}+\frac{304}{169}x+\frac{1}{169}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{49}{169} mō a, \frac{304}{169} mō b, me \frac{1}{169} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{304}{169}±\sqrt{\left(\frac{304}{169}\right)^{2}-4\left(-\frac{49}{169}\right)\times \frac{1}{169}}}{2\left(-\frac{49}{169}\right)}$$

Pūruatia \frac{304}{169} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{304}{169}±\sqrt{\frac{92416}{28561}-4\left(-\frac{49}{169}\right)\times \frac{1}{169}}}{2\left(-\frac{49}{169}\right)}$$

Whakareatia -4 ki te -\frac{49}{169}.

$$x=\frac{-\frac{304}{169}±\sqrt{\frac{92416}{28561}+\frac{196}{169}\times \frac{1}{169}}}{2\left(-\frac{49}{169}\right)}$$

Whakareatia \frac{196}{169} ki te \frac{1}{169} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\frac{304}{169}±\sqrt{\frac{92416+196}{28561}}}{2\left(-\frac{49}{169}\right)}$$

Tāpiri \frac{92416}{28561} ki te \frac{196}{28561} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\frac{304}{169}±\sqrt{\frac{548}{169}}}{2\left(-\frac{49}{169}\right)}$$

Tuhia te pūtakerua o te \frac{548}{169}.

$$x=\frac{-\frac{304}{169}±\frac{2\sqrt{137}}{13}}{2\left(-\frac{49}{169}\right)}$$

Whakareatia 2 ki te -\frac{49}{169}.

$$x=\frac{-\frac{304}{169}±\frac{2\sqrt{137}}{13}}{-\frac{98}{169}}$$

Nā, me whakaoti te whārite x=\frac{-\frac{304}{169}±\frac{2\sqrt{137}}{13}}{-\frac{98}{169}} ina he tāpiri te ±. Tāpiri -\frac{304}{169} ki te \frac{2\sqrt{137}}{13}.

$$x=\frac{\frac{2\sqrt{137}}{13}-\frac{304}{169}}{-\frac{98}{169}}$$

Whakawehe -\frac{304}{169}+\frac{2\sqrt{137}}{13} ki te -\frac{98}{169} mā te whakarea -\frac{304}{169}+\frac{2\sqrt{137}}{13} ki te tau huripoki o -\frac{98}{169}.

$$x=\frac{152-13\sqrt{137}}{49}$$

Nā, me whakaoti te whārite x=\frac{-\frac{304}{169}±\frac{2\sqrt{137}}{13}}{-\frac{98}{169}} ina he tango te ±. Tango \frac{2\sqrt{137}}{13} mai i -\frac{304}{169}.

$$x=\frac{-\frac{2\sqrt{137}}{13}-\frac{304}{169}}{-\frac{98}{169}}$$

Whakawehe -\frac{304}{169}-\frac{2\sqrt{137}}{13} ki te -\frac{98}{169} mā te whakarea -\frac{304}{169}-\frac{2\sqrt{137}}{13} ki te tau huripoki o -\frac{98}{169}.

$$x=\frac{13\sqrt{137}+152}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{152-13\sqrt{137}}{49} x=\frac{13\sqrt{137}+152}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(17x-1\right)^{2}.

$$\frac{289x^{2}-34x+1}{13^{2}}=\left(x-1\right)\times 2x$$

Tātaihia te 13 mā te pū o 2, kia riro ko 169.

$$\frac{289x^{2}-34x+1}{169}=\left(x-1\right)\times 2x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te 2.

$$\frac{289x^{2}-34x+1}{169}=\left(2x-2\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-2 ki te x.

$$\frac{289x^{2}-34x+1}{169}=2x^{2}-2x$$

Whakawehea ia wā o 289x^{2}-34x+1 ki te 169, kia riro ko \frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}.

$$\frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}=2x^{2}-2x$$

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

$$\frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}-2x^{2}=-2x$$

Pahekotia te \frac{289}{169}x^{2} me -2x^{2}, ka -\frac{49}{169}x^{2}.

$$-\frac{49}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}=-2x$$

Me tāpiri te 2x ki ngā taha e rua.

$$-\frac{49}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}+2x=0$$

Pahekotia te -\frac{34}{169}x me 2x, ka \frac{304}{169}x.

$$-\frac{49}{169}x^{2}+\frac{304}{169}x+\frac{1}{169}=0$$

Tangohia te \frac{1}{169} mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-\frac{49}{169}x^{2}+\frac{304}{169}x=-\frac{1}{169}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{49}{169}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-\frac{49}{169}x^{2}+\frac{304}{169}x}{-\frac{49}{169}}=-\frac{\frac{1}{169}}{-\frac{49}{169}}$$

Mā te whakawehe ki te -\frac{49}{169} ka wetekia te whakareanga ki te -\frac{49}{169}.

$$x^{2}+\frac{\frac{304}{169}}{-\frac{49}{169}}x=-\frac{\frac{1}{169}}{-\frac{49}{169}}$$

Whakawehe \frac{304}{169} ki te -\frac{49}{169} mā te whakarea \frac{304}{169} ki te tau huripoki o -\frac{49}{169}.

$$x^{2}-\frac{304}{49}x=-\frac{\frac{1}{169}}{-\frac{49}{169}}$$

Whakawehe -\frac{1}{169} ki te -\frac{49}{169} mā te whakarea -\frac{1}{169} ki te tau huripoki o -\frac{49}{169}.

$$x^{2}-\frac{304}{49}x=\frac{1}{49}$$

Whakawehea te -\frac{304}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{152}{49}. Nā, tāpiria te pūrua o te -\frac{152}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{304}{49}x+\left(-\frac{152}{49}\right)^{2}=\frac{1}{49}+\left(-\frac{152}{49}\right)^{2}$$

Pūruatia -\frac{152}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{304}{49}x+\frac{23104}{2401}=\frac{1}{49}+\frac{23104}{2401}$$

Tāpiri \frac{1}{49} ki te \frac{23104}{2401} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{304}{49}x+\frac{23104}{2401}=\frac{23153}{2401}$$

Tauwehea x^{2}-\frac{304}{49}x+\frac{23104}{2401}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{152}{49}\right)^{2}=\frac{23153}{2401}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{152}{49}\right)^{2}}=\sqrt{\frac{23153}{2401}}$$

Whakarūnātia.

$$x-\frac{152}{49}=\frac{13\sqrt{137}}{49} x-\frac{152}{49}=-\frac{13\sqrt{137}}{49}$$

Me tāpiri \frac{152}{49} ki ngā taha e rua o te whārite.

$$x=\frac{13\sqrt{137}+152}{49} x=\frac{152-13\sqrt{137}}{49}$$