Whakaoti i te -x^2+7x-10=0

$$- { x }^{ 2 } +7x-10 = 0$$

Whakaoti mō x

$$x=2$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=-\left(-10\right)=10$$

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 10.

$$1,10 2,5$$

Tātaihia te tapeke mō ia takirua.

$$1+10=11 2+5=7$$

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

$$a=5 b=2$$

Tuhia anō te -x^{2}+7x-10 hei \left(-x^{2}+5x\right)+\left(2x-10\right).

$$\left(-x^{2}+5x\right)+\left(2x-10\right)$$

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

$$-x\left(x-5\right)+2\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(-x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te -x+2=0.

$$x=5 x=2$$

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.

$$-x^{2}+7x-10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 7 mō b, me -10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-7±\sqrt{49+4\left(-10\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -10.

$$x=\frac{-7±\sqrt{49-40}}{2\left(-1\right)}$$

Tāpiri 49 ki te -40.

$$x=\frac{-7±\sqrt{9}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-7±3}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-7±3}{-2}$$

Nā, me whakaoti te whārite x=\frac{-7±3}{-2} ina he tāpiri te ±. Tāpiri -7 ki te 3.

$$x=-\frac{4}{-2}$$

Whakawehe -4 ki te -2.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{-7±3}{-2} ina he tango te ±. Tango 3 mai i -7.

$$x=-\frac{10}{-2}$$

Whakawehe -10 ki te -2.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=2 x=5$$

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.

$$-x^{2}+7x-10=0$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$-x^{2}+7x-10-\left(-10\right)=-\left(-10\right)$$

Mā te tango i te -10 i a ia ake anō ka toe ko te 0.

$$-x^{2}+7x=-\left(-10\right)$$

Tango -10 mai i 0.

$$-x^{2}+7x=10$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+7x}{-1}=\frac{10}{-1}$$

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

$$x^{2}+\frac{7}{-1}x=\frac{10}{-1}$$

Whakawehe 7 ki te -1.

$$x^{2}-7x=\frac{10}{-1}$$

Whakawehe 10 ki te -1.

$$x^{2}-7x=-10$$

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.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-10+\left(-\frac{7}{2}\right)^{2}$$

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

$$x^{2}-7x+\frac{49}{4}=-10+\frac{49}{4}$$

Tāpiri -10 ki te \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{9}{4}$$

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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

$$x-\frac{7}{2}=\frac{3}{2} x-\frac{7}{2}=-\frac{3}{2}$$

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

$$x=5 x=2$$