Whakaoti i te 50/49x^2-11/49x-24/49=0

$$\frac { 50 } { 49 } x ^ { 2 } - \frac { 11 } { 49 } x - \frac { 24 } { 49 } = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{4921}+11}{100}\approx 0.811498396$$

$$x=\frac{11-\sqrt{4921}}{100}\approx -0.591498396$$

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.

$$\frac{50}{49}x^{2}-\frac{11}{49}x-\frac{24}{49}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{50}{49} mō a, -\frac{11}{49} mō b, me -\frac{24}{49} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{11}{49}\right)±\sqrt{\left(-\frac{11}{49}\right)^{2}-4\times \frac{50}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

Pūruatia -\frac{11}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{11}{49}\right)±\sqrt{\frac{121}{2401}-4\times \frac{50}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

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

$$x=\frac{-\left(-\frac{11}{49}\right)±\sqrt{\frac{121}{2401}-\frac{200}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

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.

$$x=\frac{-\left(-\frac{11}{49}\right)±\sqrt{\frac{121+4800}{2401}}}{2\times \frac{50}{49}}$$

Tāpiri \frac{121}{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.

$$x=\frac{-\left(-\frac{11}{49}\right)±\sqrt{\frac{703}{343}}}{2\times \frac{50}{49}}$$

Tuhia te pūtakerua o te \frac{703}{343}.

$$x=\frac{-\left(-\frac{11}{49}\right)±\frac{\sqrt{4921}}{49}}{2\times \frac{50}{49}}$$

Ko te tauaro o -\frac{11}{49} ko \frac{11}{49}.

$$x=\frac{\frac{11}{49}±\frac{\sqrt{4921}}{49}}{2\times \frac{50}{49}}$$

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

$$x=\frac{\frac{11}{49}±\frac{\sqrt{4921}}{49}}{\frac{100}{49}}$$

Nā, me whakaoti te whārite x=\frac{\frac{11}{49}±\frac{\sqrt{4921}}{49}}{\frac{100}{49}} ina he tāpiri te ±. Tāpiri \frac{11}{49} ki te \frac{\sqrt{4921}}{49}.

$$x=\frac{\sqrt{4921}+11}{\frac{100}{49}\times 49}$$

Whakawehe \frac{11+\sqrt{4921}}{49} ki te \frac{100}{49} mā te whakarea \frac{11+\sqrt{4921}}{49} ki te tau huripoki o \frac{100}{49}.

$$x=\frac{\sqrt{4921}+11}{100}$$

Nā, me whakaoti te whārite x=\frac{\frac{11}{49}±\frac{\sqrt{4921}}{49}}{\frac{100}{49}} ina he tango te ±. Tango \frac{\sqrt{4921}}{49} mai i \frac{11}{49}.

$$x=\frac{11-\sqrt{4921}}{\frac{100}{49}\times 49}$$

Whakawehe \frac{11-\sqrt{4921}}{49} ki te \frac{100}{49} mā te whakarea \frac{11-\sqrt{4921}}{49} ki te tau huripoki o \frac{100}{49}.

$$x=\frac{11-\sqrt{4921}}{100}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{4921}+11}{100} x=\frac{11-\sqrt{4921}}{100}$$

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.

$$\frac{50}{49}x^{2}-\frac{11}{49}x-\frac{24}{49}=0$$

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

$$\frac{50}{49}x^{2}-\frac{11}{49}x-\frac{24}{49}-\left(-\frac{24}{49}\right)=-\left(-\frac{24}{49}\right)$$

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

$$\frac{50}{49}x^{2}-\frac{11}{49}x=-\left(-\frac{24}{49}\right)$$

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

$$\frac{50}{49}x^{2}-\frac{11}{49}x=\frac{24}{49}$$

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.

$$\frac{\frac{50}{49}x^{2}-\frac{11}{49}x}{\frac{50}{49}}=\frac{\frac{24}{49}}{\frac{50}{49}}$$

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

$$x^{2}+\left(-\frac{\frac{11}{49}}{\frac{50}{49}}\right)x=\frac{\frac{24}{49}}{\frac{50}{49}}$$

Whakawehe -\frac{11}{49} ki te \frac{50}{49} mā te whakarea -\frac{11}{49} ki te tau huripoki o \frac{50}{49}.

$$x^{2}-\frac{11}{50}x=\frac{\frac{24}{49}}{\frac{50}{49}}$$

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

$$x^{2}-\frac{11}{50}x=\frac{12}{25}$$

Whakawehea te -\frac{11}{50}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{100}. Nā, tāpiria te pūrua o te -\frac{11}{100} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{50}x+\left(-\frac{11}{100}\right)^{2}=\frac{12}{25}+\left(-\frac{11}{100}\right)^{2}$$

Pūruatia -\frac{11}{100} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{50}x+\frac{121}{10000}=\frac{12}{25}+\frac{121}{10000}$$

Tāpiri \frac{12}{25} ki te \frac{121}{10000} 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.

$$x^{2}-\frac{11}{50}x+\frac{121}{10000}=\frac{4921}{10000}$$

Tauwehea x^{2}-\frac{11}{50}x+\frac{121}{10000}. 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}.

$$\left(x-\frac{11}{100}\right)^{2}=\frac{4921}{10000}$$

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

$$\sqrt{\left(x-\frac{11}{100}\right)^{2}}=\sqrt{\frac{4921}{10000}}$$

Whakarūnātia.

$$x-\frac{11}{100}=\frac{\sqrt{4921}}{100} x-\frac{11}{100}=-\frac{\sqrt{4921}}{100}$$

Me tāpiri \frac{11}{100} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{4921}+11}{100} x=\frac{11-\sqrt{4921}}{100}$$