Whakaoti i te 0=left(3/4t+34-20right)left(-t+52right)

$$0 = \left( \frac{ 3 }{ 4 } t+34-20 \right) \left( -t+52 \right)$$

Whakaoti mō t

$$t=52$$

$$t = -\frac{56}{3} = -18\frac{2}{3} \approx -18.666666667$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 20 i te 34, ka 14.

$$0=\left(\frac{3}{4}t+14\right)\left(-t+52\right)$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o \frac{3}{4}t+14 ki ia tau o -t+52.

$$0=\frac{3}{4}t\left(-t\right)+\frac{3}{4}t\times 52+14\left(-t\right)+728$$

Tuhia te \frac{3}{4}\times 52 hei hautanga kotahi.

$$0=\frac{3}{4}t\left(-t\right)+\frac{3\times 52}{4}t+14\left(-t\right)+728$$

Whakareatia te 3 ki te 52, ka 156.

$$0=\frac{3}{4}t\left(-t\right)+\frac{156}{4}t+14\left(-t\right)+728$$

Whakawehea te 156 ki te 4, kia riro ko 39.

$$0=\frac{3}{4}t\left(-t\right)+39t+14\left(-t\right)+728$$

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

$$\frac{3}{4}t\left(-t\right)+39t+14\left(-t\right)+728=0$$

Whakareatia te \frac{3}{4} ki te -1, ka -\frac{3}{4}.

$$-\frac{3}{4}tt+39t+14\left(-1\right)t+728=0$$

Whakareatia te t ki te t, ka t^{2}.

$$-\frac{3}{4}t^{2}+39t+14\left(-1\right)t+728=0$$

Whakareatia te 14 ki te -1, ka -14.

$$-\frac{3}{4}t^{2}+39t-14t+728=0$$

Pahekotia te 39t me -14t, ka 25t.

$$-\frac{3}{4}t^{2}+25t+728=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{3}{4} mō a, 25 mō b, me 728 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-25±\sqrt{25^{2}-4\left(-\frac{3}{4}\right)\times 728}}{2\left(-\frac{3}{4}\right)}$$

Pūrua 25.

$$t=\frac{-25±\sqrt{625-4\left(-\frac{3}{4}\right)\times 728}}{2\left(-\frac{3}{4}\right)}$$

Whakareatia -4 ki te -\frac{3}{4}.

$$t=\frac{-25±\sqrt{625+3\times 728}}{2\left(-\frac{3}{4}\right)}$$

Whakareatia 3 ki te 728.

$$t=\frac{-25±\sqrt{625+2184}}{2\left(-\frac{3}{4}\right)}$$

Tāpiri 625 ki te 2184.

$$t=\frac{-25±\sqrt{2809}}{2\left(-\frac{3}{4}\right)}$$

Tuhia te pūtakerua o te 2809.

$$t=\frac{-25±53}{2\left(-\frac{3}{4}\right)}$$

Whakareatia 2 ki te -\frac{3}{4}.

$$t=\frac{-25±53}{-\frac{3}{2}}$$

Nā, me whakaoti te whārite t=\frac{-25±53}{-\frac{3}{2}} ina he tāpiri te ±. Tāpiri -25 ki te 53.

$$t=\frac{28}{-\frac{3}{2}}$$

Whakawehe 28 ki te -\frac{3}{2} mā te whakarea 28 ki te tau huripoki o -\frac{3}{2}.

$$t=-\frac{56}{3}$$

Nā, me whakaoti te whārite t=\frac{-25±53}{-\frac{3}{2}} ina he tango te ±. Tango 53 mai i -25.

$$t=-\frac{78}{-\frac{3}{2}}$$

Whakawehe -78 ki te -\frac{3}{2} mā te whakarea -78 ki te tau huripoki o -\frac{3}{2}.

$$t=52$$

Kua oti te whārite te whakatau.

$$t=-\frac{56}{3} t=52$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 20 i te 34, ka 14.

$$0=\left(\frac{3}{4}t+14\right)\left(-t+52\right)$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o \frac{3}{4}t+14 ki ia tau o -t+52.

$$0=\frac{3}{4}t\left(-t\right)+\frac{3}{4}t\times 52+14\left(-t\right)+728$$

Tuhia te \frac{3}{4}\times 52 hei hautanga kotahi.

$$0=\frac{3}{4}t\left(-t\right)+\frac{3\times 52}{4}t+14\left(-t\right)+728$$

Whakareatia te 3 ki te 52, ka 156.

$$0=\frac{3}{4}t\left(-t\right)+\frac{156}{4}t+14\left(-t\right)+728$$

Whakawehea te 156 ki te 4, kia riro ko 39.

$$0=\frac{3}{4}t\left(-t\right)+39t+14\left(-t\right)+728$$

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

$$\frac{3}{4}t\left(-t\right)+39t+14\left(-t\right)+728=0$$

Tangohia te 728 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$\frac{3}{4}t\left(-t\right)+39t+14\left(-t\right)=-728$$

Whakareatia te \frac{3}{4} ki te -1, ka -\frac{3}{4}.

$$-\frac{3}{4}tt+39t+14\left(-1\right)t=-728$$

Whakareatia te t ki te t, ka t^{2}.

$$-\frac{3}{4}t^{2}+39t+14\left(-1\right)t=-728$$

Whakareatia te 14 ki te -1, ka -14.

$$-\frac{3}{4}t^{2}+39t-14t=-728$$

Pahekotia te 39t me -14t, ka 25t.

$$-\frac{3}{4}t^{2}+25t=-728$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-\frac{3}{4}t^{2}+25t}{-\frac{3}{4}}=-\frac{728}{-\frac{3}{4}}$$

Mā te whakawehe ki te -\frac{3}{4} ka wetekia te whakareanga ki te -\frac{3}{4}.

$$t^{2}+\frac{25}{-\frac{3}{4}}t=-\frac{728}{-\frac{3}{4}}$$

Whakawehe 25 ki te -\frac{3}{4} mā te whakarea 25 ki te tau huripoki o -\frac{3}{4}.

$$t^{2}-\frac{100}{3}t=-\frac{728}{-\frac{3}{4}}$$

Whakawehe -728 ki te -\frac{3}{4} mā te whakarea -728 ki te tau huripoki o -\frac{3}{4}.

$$t^{2}-\frac{100}{3}t=\frac{2912}{3}$$

Whakawehea te -\frac{100}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{50}{3}. Nā, tāpiria te pūrua o te -\frac{50}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{100}{3}t+\left(-\frac{50}{3}\right)^{2}=\frac{2912}{3}+\left(-\frac{50}{3}\right)^{2}$$

Pūruatia -\frac{50}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{100}{3}t+\frac{2500}{9}=\frac{2912}{3}+\frac{2500}{9}$$

Tāpiri \frac{2912}{3} ki te \frac{2500}{9} 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.

$$t^{2}-\frac{100}{3}t+\frac{2500}{9}=\frac{11236}{9}$$

Tauwehea t^{2}-\frac{100}{3}t+\frac{2500}{9}. 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}.

$$\left(t-\frac{50}{3}\right)^{2}=\frac{11236}{9}$$

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

$$\sqrt{\left(t-\frac{50}{3}\right)^{2}}=\sqrt{\frac{11236}{9}}$$

Whakarūnātia.

$$t-\frac{50}{3}=\frac{106}{3} t-\frac{50}{3}=-\frac{106}{3}$$

Me tāpiri \frac{50}{3} ki ngā taha e rua o te whārite.

$$t=52 t=-\frac{56}{3}$$