Whakaoti i te 5=-1/60x^2+139/60x

$$5 = - \frac { 1 } { 60 } x ^ { 2 } + \frac { 139 } { 60 } x$$

Whakaoti mō x

$$x = \frac{\sqrt{18121} + 139}{2} \approx 136.807131866$$

$$x = \frac{139 - \sqrt{18121}}{2} \approx 2.192868134$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-\frac{1}{60}x^{2}+\frac{139}{60}x=5$$

Tangohia te 5 mai i ngā taha e rua.

$$-\frac{1}{60}x^{2}+\frac{139}{60}x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{60} mō a, \frac{139}{60} mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{139}{60}±\sqrt{\left(\frac{139}{60}\right)^{2}-4\left(-\frac{1}{60}\right)\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

Pūruatia \frac{139}{60} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}-4\left(-\frac{1}{60}\right)\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

Whakareatia -4 ki te -\frac{1}{60}.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}+\frac{1}{15}\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

Whakareatia \frac{1}{15} ki te -5.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}-\frac{1}{3}}}{2\left(-\frac{1}{60}\right)}$$

Tāpiri \frac{19321}{3600} ki te -\frac{1}{3} 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.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{18121}{3600}}}{2\left(-\frac{1}{60}\right)}$$

Tuhia te pūtakerua o te \frac{18121}{3600}.

$$x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{2\left(-\frac{1}{60}\right)}$$

Whakareatia 2 ki te -\frac{1}{60}.

$$x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}}$$

Nā, me whakaoti te whārite x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}} ina he tāpiri te ±. Tāpiri -\frac{139}{60} ki te \frac{\sqrt{18121}}{60}.

$$x=\frac{\sqrt{18121}-139}{-\frac{1}{30}\times 60}$$

Whakawehe \frac{-139+\sqrt{18121}}{60} ki te -\frac{1}{30} mā te whakarea \frac{-139+\sqrt{18121}}{60} ki te tau huripoki o -\frac{1}{30}.

$$x=\frac{139-\sqrt{18121}}{2}$$

Nā, me whakaoti te whārite x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}} ina he tango te ±. Tango \frac{\sqrt{18121}}{60} mai i -\frac{139}{60}.

$$x=\frac{-\sqrt{18121}-139}{-\frac{1}{30}\times 60}$$

Whakawehe \frac{-139-\sqrt{18121}}{60} ki te -\frac{1}{30} mā te whakarea \frac{-139-\sqrt{18121}}{60} ki te tau huripoki o -\frac{1}{30}.

$$x=\frac{\sqrt{18121}+139}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{139-\sqrt{18121}}{2} x=\frac{\sqrt{18121}+139}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-\frac{1}{60}x^{2}+\frac{139}{60}x=5$$

Me whakarea ngā taha e rua ki te -60.

$$\frac{-\frac{1}{60}x^{2}+\frac{139}{60}x}{-\frac{1}{60}}=\frac{5}{-\frac{1}{60}}$$

Mā te whakawehe ki te -\frac{1}{60} ka wetekia te whakareanga ki te -\frac{1}{60}.

$$x^{2}+\frac{\frac{139}{60}}{-\frac{1}{60}}x=\frac{5}{-\frac{1}{60}}$$

Whakawehe \frac{139}{60} ki te -\frac{1}{60} mā te whakarea \frac{139}{60} ki te tau huripoki o -\frac{1}{60}.

$$x^{2}-139x=\frac{5}{-\frac{1}{60}}$$

Whakawehe 5 ki te -\frac{1}{60} mā te whakarea 5 ki te tau huripoki o -\frac{1}{60}.

$$x^{2}-139x=-300$$

Whakawehea te -139, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{139}{2}. Nā, tāpiria te pūrua o te -\frac{139}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-139x+\left(-\frac{139}{2}\right)^{2}=-300+\left(-\frac{139}{2}\right)^{2}$$

Pūruatia -\frac{139}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-139x+\frac{19321}{4}=-300+\frac{19321}{4}$$

Tāpiri -300 ki te \frac{19321}{4}.

$$x^{2}-139x+\frac{19321}{4}=\frac{18121}{4}$$

Tauwehea x^{2}-139x+\frac{19321}{4}. 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}.

$$\left(x-\frac{139}{2}\right)^{2}=\frac{18121}{4}$$

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

$$\sqrt{\left(x-\frac{139}{2}\right)^{2}}=\sqrt{\frac{18121}{4}}$$

Whakarūnātia.

$$x-\frac{139}{2}=\frac{\sqrt{18121}}{2} x-\frac{139}{2}=-\frac{\sqrt{18121}}{2}$$

Me tāpiri \frac{139}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{18121}+139}{2} x=\frac{139-\sqrt{18121}}{2}$$