Whakaoti i te 5/4x^2-1/2x+025-65^2=0

$$\frac{ 5 }{ 4 } { x }^{ 2 } - \frac{ 1 }{ 2 } x+025- { 65 }^{ 2 } =0$$

Whakaoti mō x

$$x = \frac{3 \sqrt{9389} + 1}{5} \approx 58.338111424$$

$$x=\frac{1-3\sqrt{9389}}{5}\approx -57.938111424$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 0 ki te 25, ka 0.

$$\frac{5}{4}x^{2}-\frac{1}{2}x+0-65^{2}=0$$

Ko te tau i tāpiria he kore ka hua koia tonu.

$$\frac{5}{4}x^{2}-\frac{1}{2}x-65^{2}=0$$

Tātaihia te 65 mā te pū o 2, kia riro ko 4225.

$$\frac{5}{4}x^{2}-\frac{1}{2}x-4225=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{5}{4} mō a, -\frac{1}{2} mō b, me -4225 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\left(-\frac{1}{2}\right)^{2}-4\times \frac{5}{4}\left(-4225\right)}}{2\times \frac{5}{4}}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-4\times \frac{5}{4}\left(-4225\right)}}{2\times \frac{5}{4}}$$

Whakareatia -4 ki te \frac{5}{4}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-5\left(-4225\right)}}{2\times \frac{5}{4}}$$

Whakareatia -5 ki te -4225.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}+21125}}{2\times \frac{5}{4}}$$

Tāpiri \frac{1}{4} ki te 21125.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{84501}{4}}}{2\times \frac{5}{4}}$$

Tuhia te pūtakerua o te \frac{84501}{4}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\frac{3\sqrt{9389}}{2}}{2\times \frac{5}{4}}$$

Ko te tauaro o -\frac{1}{2} ko \frac{1}{2}.

$$x=\frac{\frac{1}{2}±\frac{3\sqrt{9389}}{2}}{2\times \frac{5}{4}}$$

Whakareatia 2 ki te \frac{5}{4}.

$$x=\frac{\frac{1}{2}±\frac{3\sqrt{9389}}{2}}{\frac{5}{2}}$$

Nā, me whakaoti te whārite x=\frac{\frac{1}{2}±\frac{3\sqrt{9389}}{2}}{\frac{5}{2}} ina he tāpiri te ±. Tāpiri \frac{1}{2} ki te \frac{3\sqrt{9389}}{2}.

$$x=\frac{3\sqrt{9389}+1}{2\times \frac{5}{2}}$$

Whakawehe \frac{1+3\sqrt{9389}}{2} ki te \frac{5}{2} mā te whakarea \frac{1+3\sqrt{9389}}{2} ki te tau huripoki o \frac{5}{2}.

$$x=\frac{3\sqrt{9389}+1}{5}$$

Nā, me whakaoti te whārite x=\frac{\frac{1}{2}±\frac{3\sqrt{9389}}{2}}{\frac{5}{2}} ina he tango te ±. Tango \frac{3\sqrt{9389}}{2} mai i \frac{1}{2}.

$$x=\frac{1-3\sqrt{9389}}{2\times \frac{5}{2}}$$

Whakawehe \frac{1-3\sqrt{9389}}{2} ki te \frac{5}{2} mā te whakarea \frac{1-3\sqrt{9389}}{2} ki te tau huripoki o \frac{5}{2}.

$$x=\frac{1-3\sqrt{9389}}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{3\sqrt{9389}+1}{5} x=\frac{1-3\sqrt{9389}}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 0 ki te 25, ka 0.

$$\frac{5}{4}x^{2}-\frac{1}{2}x+0-65^{2}=0$$

Ko te tau i tāpiria he kore ka hua koia tonu.

$$\frac{5}{4}x^{2}-\frac{1}{2}x-65^{2}=0$$

Tātaihia te 65 mā te pū o 2, kia riro ko 4225.

$$\frac{5}{4}x^{2}-\frac{1}{2}x-4225=0$$

Me tāpiri te 4225 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$\frac{5}{4}x^{2}-\frac{1}{2}x=4225$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{5}{4}x^{2}-\frac{1}{2}x}{\frac{5}{4}}=\frac{4225}{\frac{5}{4}}$$

Mā te whakawehe ki te \frac{5}{4} ka wetekia te whakareanga ki te \frac{5}{4}.

$$x^{2}+\left(-\frac{\frac{1}{2}}{\frac{5}{4}}\right)x=\frac{4225}{\frac{5}{4}}$$

Whakawehe -\frac{1}{2} ki te \frac{5}{4} mā te whakarea -\frac{1}{2} ki te tau huripoki o \frac{5}{4}.

$$x^{2}-\frac{2}{5}x=\frac{4225}{\frac{5}{4}}$$

Whakawehe 4225 ki te \frac{5}{4} mā te whakarea 4225 ki te tau huripoki o \frac{5}{4}.

$$x^{2}-\frac{2}{5}x=3380$$

Whakawehea te -\frac{2}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{5}. Nā, tāpiria te pūrua o te -\frac{1}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=3380+\left(-\frac{1}{5}\right)^{2}$$

Pūruatia -\frac{1}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=3380+\frac{1}{25}$$

Tāpiri 3380 ki te \frac{1}{25}.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{84501}{25}$$

Tauwehea x^{2}-\frac{2}{5}x+\frac{1}{25}. 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}.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{84501}{25}$$

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

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{84501}{25}}$$

Whakarūnātia.

$$x-\frac{1}{5}=\frac{3\sqrt{9389}}{5} x-\frac{1}{5}=-\frac{3\sqrt{9389}}{5}$$

Me tāpiri \frac{1}{5} ki ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{9389}+1}{5} x=\frac{1-3\sqrt{9389}}{5}$$