Whakaoti i te 5/4x^2-1/2x+0.25-6.5^2=0

$$\frac{ 5 }{ 4 } { x }^{ 2 } - \frac{ 1 }{ 2 } x+0.25- { 6.5 }^{ 2 } =0$$

Whakaoti mō x

$$x=-5.6$$

$$x=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 6.5 mā te pū o 2, kia riro ko 42.25.

$$\frac{5}{4}x^{2}-\frac{1}{2}x+0.25-42.25=0$$

Tangohia te 42.25 i te 0.25, ka -42.

$$\frac{5}{4}x^{2}-\frac{1}{2}x-42=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{5}{4} mō a, -\frac{1}{2} mō b, me -42 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\left(-\frac{1}{2}\right)^{2}-4\times \frac{5}{4}\left(-42\right)}}{2\times \frac{5}{4}}$$

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

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-4\times \frac{5}{4}\left(-42\right)}}{2\times \frac{5}{4}}$$

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

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-5\left(-42\right)}}{2\times \frac{5}{4}}$$

Whakareatia -5 ki te -42.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}+210}}{2\times \frac{5}{4}}$$

Tāpiri \frac{1}{4} ki te 210.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{841}{4}}}{2\times \frac{5}{4}}$$

Tuhia te pūtakerua o te \frac{841}{4}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\frac{29}{2}}{2\times \frac{5}{4}}$$

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

$$x=\frac{\frac{1}{2}±\frac{29}{2}}{2\times \frac{5}{4}}$$

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

$$x=\frac{\frac{1}{2}±\frac{29}{2}}{\frac{5}{2}}$$

Nā, me whakaoti te whārite x=\frac{\frac{1}{2}±\frac{29}{2}}{\frac{5}{2}} ina he tāpiri te ±. Tāpiri \frac{1}{2} ki te \frac{29}{2} 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.

$$x=\frac{15}{\frac{5}{2}}$$

Whakawehe 15 ki te \frac{5}{2} mā te whakarea 15 ki te tau huripoki o \frac{5}{2}.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{\frac{1}{2}±\frac{29}{2}}{\frac{5}{2}} ina he tango te ±. Tango \frac{29}{2} mai i \frac{1}{2} mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{14}{\frac{5}{2}}$$

Whakawehe -14 ki te \frac{5}{2} mā te whakarea -14 ki te tau huripoki o \frac{5}{2}.

$$x=-\frac{28}{5}$$

Kua oti te whārite te whakatau.

$$x=6 x=-\frac{28}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te 6.5 mā te pū o 2, kia riro ko 42.25.

$$\frac{5}{4}x^{2}-\frac{1}{2}x+0.25-42.25=0$$

Tangohia te 42.25 i te 0.25, ka -42.

$$\frac{5}{4}x^{2}-\frac{1}{2}x-42=0$$

Me tāpiri te 42 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$\frac{5}{4}x^{2}-\frac{1}{2}x=42$$

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.

$$\frac{\frac{5}{4}x^{2}-\frac{1}{2}x}{\frac{5}{4}}=\frac{42}{\frac{5}{4}}$$

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

$$x^{2}+\left(-\frac{\frac{1}{2}}{\frac{5}{4}}\right)x=\frac{42}{\frac{5}{4}}$$

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

$$x^{2}-\frac{2}{5}x=\frac{42}{\frac{5}{4}}$$

Whakawehe 42 ki te \frac{5}{4} mā te whakarea 42 ki te tau huripoki o \frac{5}{4}.

$$x^{2}-\frac{2}{5}x=\frac{168}{5}$$

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.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=\frac{168}{5}+\left(-\frac{1}{5}\right)^{2}$$

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

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{168}{5}+\frac{1}{25}$$

Tāpiri \frac{168}{5} ki te \frac{1}{25} 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.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{841}{25}$$

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}.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{841}{25}$$

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

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{841}{25}}$$

Whakarūnātia.

$$x-\frac{1}{5}=\frac{29}{5} x-\frac{1}{5}=-\frac{29}{5}$$

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

$$x=6 x=-\frac{28}{5}$$