Whakaoti i te 339=-q^2-2q+534

$$339 = - q ^ { 2 } - 2 q + 534$$

Whakaoti mō q

$$q=-15$$

$$q=13$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$-q^{2}-2q+534=339$$

Tangohia te 339 mai i ngā taha e rua.

$$-q^{2}-2q+534-339=0$$

Tangohia te 339 i te 534, ka 195.

$$-q^{2}-2q+195=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -q^{2}+aq+bq+195. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=-195=-195$$

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 -195.

$$1,-195 3,-65 5,-39 13,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-195=-194 3-65=-62 5-39=-34 13-15=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=13 b=-15$$

Tuhia anō te -q^{2}-2q+195 hei \left(-q^{2}+13q\right)+\left(-15q+195\right).

$$\left(-q^{2}+13q\right)+\left(-15q+195\right)$$

Tauwehea te q i te tuatahi me te 15 i te rōpū tuarua.

$$q\left(-q+13\right)+15\left(-q+13\right)$$

Whakatauwehea atu te kīanga pātahi -q+13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-q+13\right)\left(q+15\right)$$

Hei kimi otinga whārite, me whakaoti te -q+13=0 me te q+15=0.

$$q=13 q=-15$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-q^{2}-2q+534=339$$

Tangohia te 339 mai i ngā taha e rua.

$$-q^{2}-2q+534-339=0$$

Tangohia te 339 i te 534, ka 195.

$$-q^{2}-2q+195=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -2 mō b, me 195 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$q=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 195}}{2\left(-1\right)}$$

Pūrua -2.

$$q=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 195}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$q=\frac{-\left(-2\right)±\sqrt{4+4\times 195}}{2\left(-1\right)}$$

Whakareatia 4 ki te 195.

$$q=\frac{-\left(-2\right)±\sqrt{4+780}}{2\left(-1\right)}$$

Tāpiri 4 ki te 780.

$$q=\frac{-\left(-2\right)±\sqrt{784}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 784.

$$q=\frac{-\left(-2\right)±28}{2\left(-1\right)}$$

Ko te tauaro o -2 ko 2.

$$q=\frac{2±28}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$q=\frac{2±28}{-2}$$

Nā, me whakaoti te whārite q=\frac{2±28}{-2} ina he tāpiri te ±. Tāpiri 2 ki te 28.

$$q=\frac{30}{-2}$$

Whakawehe 30 ki te -2.

$$q=-15$$

Nā, me whakaoti te whārite q=\frac{2±28}{-2} ina he tango te ±. Tango 28 mai i 2.

$$q=-\frac{26}{-2}$$

Whakawehe -26 ki te -2.

$$q=13$$

Kua oti te whārite te whakatau.

$$q=-15 q=13$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-q^{2}-2q+534=339$$

Tangohia te 534 mai i ngā taha e rua.

$$-q^{2}-2q=339-534$$

Tangohia te 534 i te 339, ka -195.

$$-q^{2}-2q=-195$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-q^{2}-2q}{-1}=-\frac{195}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$q^{2}+\left(-\frac{2}{-1}\right)q=-\frac{195}{-1}$$

Whakawehe -2 ki te -1.

$$q^{2}+2q=-\frac{195}{-1}$$

Whakawehe -195 ki te -1.

$$q^{2}+2q=195$$

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$q^{2}+2q+1^{2}=195+1^{2}$$

Pūrua 1.

$$q^{2}+2q+1=195+1$$

Tāpiri 195 ki te 1.

$$q^{2}+2q+1=196$$

Tauwehea q^{2}+2q+1. 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}.

$$\left(q+1\right)^{2}=196$$

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

$$\sqrt{\left(q+1\right)^{2}}=\sqrt{196}$$

Whakarūnātia.

$$q+1=14 q+1=-14$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$q=13 q=-15$$