Whakaoti i te x=-16x^2+81x+5

$$x=-16 { x }^{ 2 } +81x+5$$

Whakaoti mō x

$$x=\frac{\sqrt{105}}{4}+\frac{5}{2}\approx 5.061737691$$

$$x=-\frac{\sqrt{105}}{4}+\frac{5}{2}\approx -0.061737691$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 16x^{2} ki ngā taha e rua.

$$x+16x^{2}=81x+5$$

Tangohia te 81x mai i ngā taha e rua.

$$x+16x^{2}-81x=5$$

Pahekotia te x me -81x, ka -80x.

$$-80x+16x^{2}=5$$

Tangohia te 5 mai i ngā taha e rua.

$$-80x+16x^{2}-5=0$$

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.

$$16x^{2}-80x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, -80 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 16\left(-5\right)}}{2\times 16}$$

Pūrua -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 16\left(-5\right)}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-\left(-80\right)±\sqrt{6400-64\left(-5\right)}}{2\times 16}$$

Whakareatia -64 ki te -5.

$$x=\frac{-\left(-80\right)±\sqrt{6400+320}}{2\times 16}$$

Tāpiri 6400 ki te 320.

$$x=\frac{-\left(-80\right)±\sqrt{6720}}{2\times 16}$$

Tuhia te pūtakerua o te 6720.

$$x=\frac{-\left(-80\right)±8\sqrt{105}}{2\times 16}$$

Ko te tauaro o -80 ko 80.

$$x=\frac{80±8\sqrt{105}}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{80±8\sqrt{105}}{32}$$

Nā, me whakaoti te whārite x=\frac{80±8\sqrt{105}}{32} ina he tāpiri te ±. Tāpiri 80 ki te 8\sqrt{105}.

$$x=\frac{8\sqrt{105}+80}{32}$$

Whakawehe 80+8\sqrt{105} ki te 32.

$$x=\frac{\sqrt{105}}{4}+\frac{5}{2}$$

Nā, me whakaoti te whārite x=\frac{80±8\sqrt{105}}{32} ina he tango te ±. Tango 8\sqrt{105} mai i 80.

$$x=\frac{80-8\sqrt{105}}{32}$$

Whakawehe 80-8\sqrt{105} ki te 32.

$$x=-\frac{\sqrt{105}}{4}+\frac{5}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{105}}{4}+\frac{5}{2} x=-\frac{\sqrt{105}}{4}+\frac{5}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 16x^{2} ki ngā taha e rua.

$$x+16x^{2}=81x+5$$

Tangohia te 81x mai i ngā taha e rua.

$$x+16x^{2}-81x=5$$

Pahekotia te x me -81x, ka -80x.

$$-80x+16x^{2}=5$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$16x^{2}-80x=5$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}-80x}{16}=\frac{5}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$x^{2}+\left(-\frac{80}{16}\right)x=\frac{5}{16}$$

Whakawehe -80 ki te 16.

$$x^{2}-5x=\frac{5}{16}$$

Whakawehea te -5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{2}. Nā, tāpiria te pūrua o te -\frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=\frac{5}{16}+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=\frac{5}{16}+\frac{25}{4}$$

Tāpiri \frac{5}{16} ki te \frac{25}{4} 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.

$$x^{2}-5x+\frac{25}{4}=\frac{105}{16}$$

Tauwehea x^{2}-5x+\frac{25}{4}. 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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{105}{16}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{105}{16}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{\sqrt{105}}{4} x-\frac{5}{2}=-\frac{\sqrt{105}}{4}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{105}}{4}+\frac{5}{2} x=-\frac{\sqrt{105}}{4}+\frac{5}{2}$$