Whakaoti i te 4x*(4x-21)=80

$$4 x \cdot ( 4 x - 21 ) = 80$$

Whakaoti mō x

$$x = \frac{\sqrt{761} + 21}{8} \approx 6.073278556$$

$$x=\frac{21-\sqrt{761}}{8}\approx -0.823278556$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 4x ki te 4x-21.

$$16x^{2}-84x=80$$

Tangohia te 80 mai i ngā taha e rua.

$$16x^{2}-84x-80=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, -84 mō b, me -80 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-84\right)±\sqrt{\left(-84\right)^{2}-4\times 16\left(-80\right)}}{2\times 16}$$

Pūrua -84.

$$x=\frac{-\left(-84\right)±\sqrt{7056-4\times 16\left(-80\right)}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-\left(-84\right)±\sqrt{7056-64\left(-80\right)}}{2\times 16}$$

Whakareatia -64 ki te -80.

$$x=\frac{-\left(-84\right)±\sqrt{7056+5120}}{2\times 16}$$

Tāpiri 7056 ki te 5120.

$$x=\frac{-\left(-84\right)±\sqrt{12176}}{2\times 16}$$

Tuhia te pūtakerua o te 12176.

$$x=\frac{-\left(-84\right)±4\sqrt{761}}{2\times 16}$$

Ko te tauaro o -84 ko 84.

$$x=\frac{84±4\sqrt{761}}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{84±4\sqrt{761}}{32}$$

Nā, me whakaoti te whārite x=\frac{84±4\sqrt{761}}{32} ina he tāpiri te ±. Tāpiri 84 ki te 4\sqrt{761}.

$$x=\frac{4\sqrt{761}+84}{32}$$

Whakawehe 84+4\sqrt{761} ki te 32.

$$x=\frac{\sqrt{761}+21}{8}$$

Nā, me whakaoti te whārite x=\frac{84±4\sqrt{761}}{32} ina he tango te ±. Tango 4\sqrt{761} mai i 84.

$$x=\frac{84-4\sqrt{761}}{32}$$

Whakawehe 84-4\sqrt{761} ki te 32.

$$x=\frac{21-\sqrt{761}}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{761}+21}{8} x=\frac{21-\sqrt{761}}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 4x ki te 4x-21.

$$16x^{2}-84x=80$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}-84x}{16}=\frac{80}{16}$$

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

$$x^{2}+\left(-\frac{84}{16}\right)x=\frac{80}{16}$$

Whakahekea te hautanga \frac{-84}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{21}{4}x=\frac{80}{16}$$

Whakawehe 80 ki te 16.

$$x^{2}-\frac{21}{4}x=5$$

Whakawehea te -\frac{21}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{21}{8}. Nā, tāpiria te pūrua o te -\frac{21}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{21}{4}x+\left(-\frac{21}{8}\right)^{2}=5+\left(-\frac{21}{8}\right)^{2}$$

Pūruatia -\frac{21}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{21}{4}x+\frac{441}{64}=5+\frac{441}{64}$$

Tāpiri 5 ki te \frac{441}{64}.

$$x^{2}-\frac{21}{4}x+\frac{441}{64}=\frac{761}{64}$$

Tauwehea x^{2}-\frac{21}{4}x+\frac{441}{64}. 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}.

$$\left(x-\frac{21}{8}\right)^{2}=\frac{761}{64}$$

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

$$\sqrt{\left(x-\frac{21}{8}\right)^{2}}=\sqrt{\frac{761}{64}}$$

Whakarūnātia.

$$x-\frac{21}{8}=\frac{\sqrt{761}}{8} x-\frac{21}{8}=-\frac{\sqrt{761}}{8}$$

Me tāpiri \frac{21}{8} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{761}+21}{8} x=\frac{21-\sqrt{761}}{8}$$