Whakaoti i te 5x^2-35x-90=0

$$5 x ^ { 2 } - 35 x - 90 = 0$$

Whakaoti mō x

$$x=-2$$

$$x=9$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 5.

$$x^{2}-7x-18=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=1\left(-18\right)=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-18=-17 2-9=-7 3-6=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-9 b=2$$

Tuhia anō te x^{2}-7x-18 hei \left(x^{2}-9x\right)+\left(2x-18\right).

$$\left(x^{2}-9x\right)+\left(2x-18\right)$$

Tauwehea te x i te tuatahi me te 2 i te rōpū tuarua.

$$x\left(x-9\right)+2\left(x-9\right)$$

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

$$\left(x-9\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+2=0.

$$x=9 x=-2$$

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.

$$5x^{2}-35x-90=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -35 mō b, me -90 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}-4\times 5\left(-90\right)}}{2\times 5}$$

Pūrua -35.

$$x=\frac{-\left(-35\right)±\sqrt{1225-4\times 5\left(-90\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-35\right)±\sqrt{1225-20\left(-90\right)}}{2\times 5}$$

Whakareatia -20 ki te -90.

$$x=\frac{-\left(-35\right)±\sqrt{1225+1800}}{2\times 5}$$

Tāpiri 1225 ki te 1800.

$$x=\frac{-\left(-35\right)±\sqrt{3025}}{2\times 5}$$

Tuhia te pūtakerua o te 3025.

$$x=\frac{-\left(-35\right)±55}{2\times 5}$$

Ko te tauaro o -35 ko 35.

$$x=\frac{35±55}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{35±55}{10}$$

Nā, me whakaoti te whārite x=\frac{35±55}{10} ina he tāpiri te ±. Tāpiri 35 ki te 55.

$$x=\frac{90}{10}$$

Whakawehe 90 ki te 10.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{35±55}{10} ina he tango te ±. Tango 55 mai i 35.

$$x=-\frac{20}{10}$$

Whakawehe -20 ki te 10.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=9 x=-2$$

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.

$$5x^{2}-35x-90=0$$

Me tāpiri 90 ki ngā taha e rua o te whārite.

$$5x^{2}-35x-90-\left(-90\right)=-\left(-90\right)$$

Mā te tango i te -90 i a ia ake anō ka toe ko te 0.

$$5x^{2}-35x=-\left(-90\right)$$

Tango -90 mai i 0.

$$5x^{2}-35x=90$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-35x}{5}=\frac{90}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\left(-\frac{35}{5}\right)x=\frac{90}{5}$$

Whakawehe -35 ki te 5.

$$x^{2}-7x=\frac{90}{5}$$

Whakawehe 90 ki te 5.

$$x^{2}-7x=18$$

Whakawehea te -7, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{2}. Nā, tāpiria te pūrua o te -\frac{7}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=18+\left(-\frac{7}{2}\right)^{2}$$

Pūruatia -\frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-7x+\frac{49}{4}=18+\frac{49}{4}$$

Tāpiri 18 ki te \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{121}{4}$$

Tauwehea x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x-\frac{7}{2}=\frac{11}{2} x-\frac{7}{2}=-\frac{11}{2}$$

Me tāpiri \frac{7}{2} ki ngā taha e rua o te whārite.

$$x=9 x=-2$$