Whakaoti i te 50/49x^2-10/49x-24/49=0

$$\frac{ 50 }{ 49 } { x }^{ 2 } - \frac{ 10 }{ 49 } x- \frac{ 24 }{ 49 } = 0$$

Whakaoti mō x

$$x=-\frac{3}{5}=-0.6$$

$$x=\frac{4}{5}=0.8$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$\frac{50}{49}x^{2}-\frac{10}{49}x-\frac{24}{49}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{50}{49} mō a, -\frac{10}{49} mō b, me -\frac{24}{49} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\left(-\frac{10}{49}\right)^{2}-4\times \frac{50}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

Pūruatia -\frac{10}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\frac{100}{2401}-4\times \frac{50}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

Whakareatia -4 ki te \frac{50}{49}.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\frac{100}{2401}-\frac{200}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

Whakareatia -\frac{200}{49} ki te -\frac{24}{49} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\frac{100+4800}{2401}}}{2\times \frac{50}{49}}$$

Tāpiri \frac{100}{2401} ki te \frac{4800}{2401} 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.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\frac{100}{49}}}{2\times \frac{50}{49}}$$

Tuhia te pūtakerua o te \frac{100}{49}.

$$x=\frac{-\left(-\frac{10}{49}\right)±\frac{10}{7}}{2\times \frac{50}{49}}$$

Ko te tauaro o -\frac{10}{49} ko \frac{10}{49}.

$$x=\frac{\frac{10}{49}±\frac{10}{7}}{2\times \frac{50}{49}}$$

Whakareatia 2 ki te \frac{50}{49}.

$$x=\frac{\frac{10}{49}±\frac{10}{7}}{\frac{100}{49}}$$

Nā, me whakaoti te whārite x=\frac{\frac{10}{49}±\frac{10}{7}}{\frac{100}{49}} ina he tāpiri te ±. Tāpiri \frac{10}{49} ki te \frac{10}{7} 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.

$$x=\frac{\frac{80}{49}}{\frac{100}{49}}$$

Whakawehe \frac{80}{49} ki te \frac{100}{49} mā te whakarea \frac{80}{49} ki te tau huripoki o \frac{100}{49}.

$$x=\frac{4}{5}$$

Nā, me whakaoti te whārite x=\frac{\frac{10}{49}±\frac{10}{7}}{\frac{100}{49}} ina he tango te ±. Tango \frac{10}{7} mai i \frac{10}{49} 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.

$$x=-\frac{\frac{60}{49}}{\frac{100}{49}}$$

Whakawehe -\frac{60}{49} ki te \frac{100}{49} mā te whakarea -\frac{60}{49} ki te tau huripoki o \frac{100}{49}.

$$x=-\frac{3}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{4}{5} x=-\frac{3}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$\frac{50}{49}x^{2}-\frac{10}{49}x-\frac{24}{49}=0$$

Me tāpiri \frac{24}{49} ki ngā taha e rua o te whārite.

$$\frac{50}{49}x^{2}-\frac{10}{49}x-\frac{24}{49}-\left(-\frac{24}{49}\right)=-\left(-\frac{24}{49}\right)$$

Mā te tango i te -\frac{24}{49} i a ia ake anō ka toe ko te 0.

$$\frac{50}{49}x^{2}-\frac{10}{49}x=-\left(-\frac{24}{49}\right)$$

Tango -\frac{24}{49} mai i 0.

$$\frac{50}{49}x^{2}-\frac{10}{49}x=\frac{24}{49}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{50}{49}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{50}{49}x^{2}-\frac{10}{49}x}{\frac{50}{49}}=\frac{\frac{24}{49}}{\frac{50}{49}}$$

Mā te whakawehe ki te \frac{50}{49} ka wetekia te whakareanga ki te \frac{50}{49}.

$$x^{2}+\left(-\frac{\frac{10}{49}}{\frac{50}{49}}\right)x=\frac{\frac{24}{49}}{\frac{50}{49}}$$

Whakawehe -\frac{10}{49} ki te \frac{50}{49} mā te whakarea -\frac{10}{49} ki te tau huripoki o \frac{50}{49}.

$$x^{2}-\frac{1}{5}x=\frac{\frac{24}{49}}{\frac{50}{49}}$$

Whakawehe \frac{24}{49} ki te \frac{50}{49} mā te whakarea \frac{24}{49} ki te tau huripoki o \frac{50}{49}.

$$x^{2}-\frac{1}{5}x=\frac{12}{25}$$

Whakawehea te -\frac{1}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{10}. Nā, tāpiria te pūrua o te -\frac{1}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{5}x+\left(-\frac{1}{10}\right)^{2}=\frac{12}{25}+\left(-\frac{1}{10}\right)^{2}$$

Pūruatia -\frac{1}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=\frac{12}{25}+\frac{1}{100}$$

Tāpiri \frac{12}{25} ki te \frac{1}{100} 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.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=\frac{49}{100}$$

Tauwehea x^{2}-\frac{1}{5}x+\frac{1}{100}. 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}.

$$\left(x-\frac{1}{10}\right)^{2}=\frac{49}{100}$$

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

$$\sqrt{\left(x-\frac{1}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Whakarūnātia.

$$x-\frac{1}{10}=\frac{7}{10} x-\frac{1}{10}=-\frac{7}{10}$$

Me tāpiri \frac{1}{10} ki ngā taha e rua o te whārite.

$$x=\frac{4}{5} x=-\frac{3}{5}$$