Whakaoti i te 15*(1-x)*(1+x)+7x-3=0

$$15 \times (1-x) \times (1+x)+7x-3=0$$

Whakaoti mō x

$$x = \frac{\sqrt{769} + 7}{30} \approx 1.157694975$$

$$x=\frac{7-\sqrt{769}}{30}\approx -0.691028308$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 15 ki te 1-x.

$$\left(15-15x\right)\left(1+x\right)+7x-3=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 15-15x ki te 1+x ka whakakotahi i ngā kupu rite.

$$15-15x^{2}+7x-3=0$$

Tangohia te 3 i te 15, ka 12.

$$12-15x^{2}+7x=0$$

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.

$$-15x^{2}+7x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -15 mō a, 7 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-15\right)\times 12}}{2\left(-15\right)}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-15\right)\times 12}}{2\left(-15\right)}$$

Whakareatia -4 ki te -15.

$$x=\frac{-7±\sqrt{49+60\times 12}}{2\left(-15\right)}$$

Whakareatia 60 ki te 12.

$$x=\frac{-7±\sqrt{49+720}}{2\left(-15\right)}$$

Tāpiri 49 ki te 720.

$$x=\frac{-7±\sqrt{769}}{2\left(-15\right)}$$

Whakareatia 2 ki te -15.

$$x=\frac{-7±\sqrt{769}}{-30}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{769}}{-30} ina he tāpiri te ±. Tāpiri -7 ki te \sqrt{769}.

$$x=\frac{\sqrt{769}-7}{-30}$$

Whakawehe -7+\sqrt{769} ki te -30.

$$x=\frac{7-\sqrt{769}}{30}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{769}}{-30} ina he tango te ±. Tango \sqrt{769} mai i -7.

$$x=\frac{-\sqrt{769}-7}{-30}$$

Whakawehe -7-\sqrt{769} ki te -30.

$$x=\frac{\sqrt{769}+7}{30}$$

Kua oti te whārite te whakatau.

$$x=\frac{7-\sqrt{769}}{30} x=\frac{\sqrt{769}+7}{30}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 15 ki te 1-x.

$$\left(15-15x\right)\left(1+x\right)+7x-3=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 15-15x ki te 1+x ka whakakotahi i ngā kupu rite.

$$15-15x^{2}+7x-3=0$$

Tangohia te 3 i te 15, ka 12.

$$12-15x^{2}+7x=0$$

Tangohia te 12 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-15x^{2}+7x=-12$$

Whakawehea ngā taha e rua ki te -15.

$$\frac{-15x^{2}+7x}{-15}=-\frac{12}{-15}$$

Mā te whakawehe ki te -15 ka wetekia te whakareanga ki te -15.

$$x^{2}+\frac{7}{-15}x=-\frac{12}{-15}$$

Whakawehe 7 ki te -15.

$$x^{2}-\frac{7}{15}x=-\frac{12}{-15}$$

Whakahekea te hautanga \frac{-12}{-15} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{7}{15}x=\frac{4}{5}$$

Whakawehea te -\frac{7}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{30}. Nā, tāpiria te pūrua o te -\frac{7}{30} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{15}x+\left(-\frac{7}{30}\right)^{2}=\frac{4}{5}+\left(-\frac{7}{30}\right)^{2}$$

Pūruatia -\frac{7}{30} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{15}x+\frac{49}{900}=\frac{4}{5}+\frac{49}{900}$$

Tāpiri \frac{4}{5} ki te \frac{49}{900} 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.

$$x^{2}-\frac{7}{15}x+\frac{49}{900}=\frac{769}{900}$$

Tauwehea x^{2}-\frac{7}{15}x+\frac{49}{900}. 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}.

$$\left(x-\frac{7}{30}\right)^{2}=\frac{769}{900}$$

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

$$\sqrt{\left(x-\frac{7}{30}\right)^{2}}=\sqrt{\frac{769}{900}}$$

Whakarūnātia.

$$x-\frac{7}{30}=\frac{\sqrt{769}}{30} x-\frac{7}{30}=-\frac{\sqrt{769}}{30}$$

Me tāpiri \frac{7}{30} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{769}+7}{30} x=\frac{7-\sqrt{769}}{30}$$