Whakaoti i te 20=-49t^2+20t+130

$$20 = - 49 t ^ { 2 } + 20 t + 130$$

Whakaoti mō t

$$t = \frac{3 \sqrt{610} + 10}{49} \approx 1.716214984$$

$$t=\frac{10-3\sqrt{610}}{49}\approx -1.308051719$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-49t^{2}+20t+130=20$$

Tangohia te 20 mai i ngā taha e rua.

$$-49t^{2}+20t+130-20=0$$

Tangohia te 20 i te 130, ka 110.

$$-49t^{2}+20t+110=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -49 mō a, 20 mō b, me 110 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\left(-49\right)\times 110}}{2\left(-49\right)}$$

Pūrua 20.

$$t=\frac{-20±\sqrt{400-4\left(-49\right)\times 110}}{2\left(-49\right)}$$

Whakareatia -4 ki te -49.

$$t=\frac{-20±\sqrt{400+196\times 110}}{2\left(-49\right)}$$

Whakareatia 196 ki te 110.

$$t=\frac{-20±\sqrt{400+21560}}{2\left(-49\right)}$$

Tāpiri 400 ki te 21560.

$$t=\frac{-20±\sqrt{21960}}{2\left(-49\right)}$$

Tuhia te pūtakerua o te 21960.

$$t=\frac{-20±6\sqrt{610}}{2\left(-49\right)}$$

Whakareatia 2 ki te -49.

$$t=\frac{-20±6\sqrt{610}}{-98}$$

Nā, me whakaoti te whārite t=\frac{-20±6\sqrt{610}}{-98} ina he tāpiri te ±. Tāpiri -20 ki te 6\sqrt{610}.

$$t=\frac{6\sqrt{610}-20}{-98}$$

Whakawehe -20+6\sqrt{610} ki te -98.

$$t=\frac{10-3\sqrt{610}}{49}$$

Nā, me whakaoti te whārite t=\frac{-20±6\sqrt{610}}{-98} ina he tango te ±. Tango 6\sqrt{610} mai i -20.

$$t=\frac{-6\sqrt{610}-20}{-98}$$

Whakawehe -20-6\sqrt{610} ki te -98.

$$t=\frac{3\sqrt{610}+10}{49}$$

Kua oti te whārite te whakatau.

$$t=\frac{10-3\sqrt{610}}{49} t=\frac{3\sqrt{610}+10}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-49t^{2}+20t+130=20$$

Tangohia te 130 mai i ngā taha e rua.

$$-49t^{2}+20t=20-130$$

Tangohia te 130 i te 20, ka -110.

$$-49t^{2}+20t=-110$$

Whakawehea ngā taha e rua ki te -49.

$$\frac{-49t^{2}+20t}{-49}=-\frac{110}{-49}$$

Mā te whakawehe ki te -49 ka wetekia te whakareanga ki te -49.

$$t^{2}+\frac{20}{-49}t=-\frac{110}{-49}$$

Whakawehe 20 ki te -49.

$$t^{2}-\frac{20}{49}t=-\frac{110}{-49}$$

Whakawehe -110 ki te -49.

$$t^{2}-\frac{20}{49}t=\frac{110}{49}$$

Whakawehea te -\frac{20}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{10}{49}. Nā, tāpiria te pūrua o te -\frac{10}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{20}{49}t+\left(-\frac{10}{49}\right)^{2}=\frac{110}{49}+\left(-\frac{10}{49}\right)^{2}$$

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

$$t^{2}-\frac{20}{49}t+\frac{100}{2401}=\frac{110}{49}+\frac{100}{2401}$$

Tāpiri \frac{110}{49} ki te \frac{100}{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.

$$t^{2}-\frac{20}{49}t+\frac{100}{2401}=\frac{5490}{2401}$$

Tauwehea t^{2}-\frac{20}{49}t+\frac{100}{2401}. 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}.

$$\left(t-\frac{10}{49}\right)^{2}=\frac{5490}{2401}$$

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

$$\sqrt{\left(t-\frac{10}{49}\right)^{2}}=\sqrt{\frac{5490}{2401}}$$

Whakarūnātia.

$$t-\frac{10}{49}=\frac{3\sqrt{610}}{49} t-\frac{10}{49}=-\frac{3\sqrt{610}}{49}$$

Me tāpiri \frac{10}{49} ki ngā taha e rua o te whārite.

$$t=\frac{3\sqrt{610}+10}{49} t=\frac{10-3\sqrt{610}}{49}$$