Whakaoti i te 20t^2-17t=63

$$20 t ^ { 2 } - 17 t = 63$$

Whakaoti mō t

$$t = -\frac{7}{5} = -1\frac{2}{5} = -1.4$$

$$t = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 63 mai i ngā taha e rua.

$$20t^{2}-17t-63=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 20t^{2}+at+bt-63. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-17 ab=20\left(-63\right)=-1260$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -1260.

$$1,-1260 2,-630 3,-420 4,-315 5,-252 6,-210 7,-180 9,-140 10,-126 12,-105 14,-90 15,-84 18,-70 20,-63 21,-60 28,-45 30,-42 35,-36$$

Tātaihia te tapeke mō ia takirua.

$$1-1260=-1259 2-630=-628 3-420=-417 4-315=-311 5-252=-247 6-210=-204 7-180=-173 9-140=-131 10-126=-116 12-105=-93 14-90=-76 15-84=-69 18-70=-52 20-63=-43 21-60=-39 28-45=-17 30-42=-12 35-36=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -17.

$$a=-45 b=28$$

Tuhia anō te 20t^{2}-17t-63 hei \left(20t^{2}-45t\right)+\left(28t-63\right).

$$\left(20t^{2}-45t\right)+\left(28t-63\right)$$

Tauwehea te 5t i te tuatahi me te 7 i te rōpū tuarua.

$$5t\left(4t-9\right)+7\left(4t-9\right)$$

Whakatauwehea atu te kīanga pātahi 4t-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4t-9\right)\left(5t+7\right)$$

Hei kimi otinga whārite, me whakaoti te 4t-9=0 me te 5t+7=0.

$$t=\frac{9}{4} t=-\frac{7}{5}$$

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.

$$20t^{2}-17t=63$$

Me tango 63 mai i ngā taha e rua o te whārite.

$$20t^{2}-17t-63=63-63$$

Mā te tango i te 63 i a ia ake anō ka toe ko te 0.

$$20t^{2}-17t-63=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 20 mō a, -17 mō b, me -63 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 20\left(-63\right)}}{2\times 20}$$

Pūrua -17.

$$t=\frac{-\left(-17\right)±\sqrt{289-4\times 20\left(-63\right)}}{2\times 20}$$

Whakareatia -4 ki te 20.

$$t=\frac{-\left(-17\right)±\sqrt{289-80\left(-63\right)}}{2\times 20}$$

Whakareatia -80 ki te -63.

$$t=\frac{-\left(-17\right)±\sqrt{289+5040}}{2\times 20}$$

Tāpiri 289 ki te 5040.

$$t=\frac{-\left(-17\right)±\sqrt{5329}}{2\times 20}$$

Tuhia te pūtakerua o te 5329.

$$t=\frac{-\left(-17\right)±73}{2\times 20}$$

Ko te tauaro o -17 ko 17.

$$t=\frac{17±73}{2\times 20}$$

Whakareatia 2 ki te 20.

$$t=\frac{17±73}{40}$$

Nā, me whakaoti te whārite t=\frac{17±73}{40} ina he tāpiri te ±. Tāpiri 17 ki te 73.

$$t=\frac{90}{40}$$

Whakahekea te hautanga \frac{90}{40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$t=\frac{9}{4}$$

Nā, me whakaoti te whārite t=\frac{17±73}{40} ina he tango te ±. Tango 73 mai i 17.

$$t=-\frac{56}{40}$$

Whakahekea te hautanga \frac{-56}{40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$t=-\frac{7}{5}$$

Kua oti te whārite te whakatau.

$$t=\frac{9}{4} t=-\frac{7}{5}$$

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.

$$20t^{2}-17t=63$$

Whakawehea ngā taha e rua ki te 20.

$$\frac{20t^{2}-17t}{20}=\frac{63}{20}$$

Mā te whakawehe ki te 20 ka wetekia te whakareanga ki te 20.

$$t^{2}-\frac{17}{20}t=\frac{63}{20}$$

Whakawehea te -\frac{17}{20}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{17}{40}. Nā, tāpiria te pūrua o te -\frac{17}{40} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{17}{20}t+\left(-\frac{17}{40}\right)^{2}=\frac{63}{20}+\left(-\frac{17}{40}\right)^{2}$$

Pūruatia -\frac{17}{40} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{17}{20}t+\frac{289}{1600}=\frac{63}{20}+\frac{289}{1600}$$

Tāpiri \frac{63}{20} ki te \frac{289}{1600} 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.

$$t^{2}-\frac{17}{20}t+\frac{289}{1600}=\frac{5329}{1600}$$

Tauwehea t^{2}-\frac{17}{20}t+\frac{289}{1600}. 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}.

$$\left(t-\frac{17}{40}\right)^{2}=\frac{5329}{1600}$$

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

$$\sqrt{\left(t-\frac{17}{40}\right)^{2}}=\sqrt{\frac{5329}{1600}}$$

Whakarūnātia.

$$t-\frac{17}{40}=\frac{73}{40} t-\frac{17}{40}=-\frac{73}{40}$$

Me tāpiri \frac{17}{40} ki ngā taha e rua o te whārite.

$$t=\frac{9}{4} t=-\frac{7}{5}$$