Whakaoti i te -x^2+45x-200=124

$$- { x }^{ 2 } +45x-200=124$$

Whakaoti mō x

$$x=9$$

$$x=36$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 124 mai i ngā taha e rua.

$$-x^{2}+45x-200-124=0$$

Tangohia te 124 i te -200, ka -324.

$$-x^{2}+45x-324=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-324. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=45 ab=-\left(-324\right)=324$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 324.

$$1,324 2,162 3,108 4,81 6,54 9,36 12,27 18,18$$

Tātaihia te tapeke mō ia takirua.

$$1+324=325 2+162=164 3+108=111 4+81=85 6+54=60 9+36=45 12+27=39 18+18=36$$

Ko te otinga te takirua ka hoatu i te tapeke 45.

$$a=36 b=9$$

Tuhia anō te -x^{2}+45x-324 hei \left(-x^{2}+36x\right)+\left(9x-324\right).

$$\left(-x^{2}+36x\right)+\left(9x-324\right)$$

Tauwehea te -x i te tuatahi me te 9 i te rōpū tuarua.

$$-x\left(x-36\right)+9\left(x-36\right)$$

Whakatauwehea atu te kīanga pātahi x-36 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-36\right)\left(-x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x-36=0 me te -x+9=0.

$$x=36 x=9$$

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.

$$-x^{2}+45x-200=124$$

Me tango 124 mai i ngā taha e rua o te whārite.

$$-x^{2}+45x-200-124=124-124$$

Mā te tango i te 124 i a ia ake anō ka toe ko te 0.

$$-x^{2}+45x-200-124=0$$

Tango 124 mai i -200.

$$-x^{2}+45x-324=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 45 mō b, me -324 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}-4\left(-1\right)\left(-324\right)}}{2\left(-1\right)}$$

Pūrua 45.

$$x=\frac{-45±\sqrt{2025-4\left(-1\right)\left(-324\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-45±\sqrt{2025+4\left(-324\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -324.

$$x=\frac{-45±\sqrt{2025-1296}}{2\left(-1\right)}$$

Tāpiri 2025 ki te -1296.

$$x=\frac{-45±\sqrt{729}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-45±27}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-45±27}{-2}$$

Nā, me whakaoti te whārite x=\frac{-45±27}{-2} ina he tāpiri te ±. Tāpiri -45 ki te 27.

$$x=-\frac{18}{-2}$$

Whakawehe -18 ki te -2.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{-45±27}{-2} ina he tango te ±. Tango 27 mai i -45.

$$x=-\frac{72}{-2}$$

Whakawehe -72 ki te -2.

$$x=36$$

Kua oti te whārite te whakatau.

$$x=9 x=36$$

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.

$$-x^{2}+45x-200=124$$

Me tāpiri 200 ki ngā taha e rua o te whārite.

$$-x^{2}+45x-200-\left(-200\right)=124-\left(-200\right)$$

Mā te tango i te -200 i a ia ake anō ka toe ko te 0.

$$-x^{2}+45x=124-\left(-200\right)$$

Tango -200 mai i 124.

$$-x^{2}+45x=324$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+45x}{-1}=\frac{324}{-1}$$

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

$$x^{2}+\frac{45}{-1}x=\frac{324}{-1}$$

Whakawehe 45 ki te -1.

$$x^{2}-45x=\frac{324}{-1}$$

Whakawehe 324 ki te -1.

$$x^{2}-45x=-324$$

Whakawehea te -45, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{45}{2}. Nā, tāpiria te pūrua o te -\frac{45}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-45x+\left(-\frac{45}{2}\right)^{2}=-324+\left(-\frac{45}{2}\right)^{2}$$

Pūruatia -\frac{45}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-45x+\frac{2025}{4}=-324+\frac{2025}{4}$$

Tāpiri -324 ki te \frac{2025}{4}.

$$x^{2}-45x+\frac{2025}{4}=\frac{729}{4}$$

Tauwehea x^{2}-45x+\frac{2025}{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}.

$$\left(x-\frac{45}{2}\right)^{2}=\frac{729}{4}$$

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

$$\sqrt{\left(x-\frac{45}{2}\right)^{2}}=\sqrt{\frac{729}{4}}$$

Whakarūnātia.

$$x-\frac{45}{2}=\frac{27}{2} x-\frac{45}{2}=-\frac{27}{2}$$

Me tāpiri \frac{45}{2} ki ngā taha e rua o te whārite.

$$x=36 x=9$$