Whakaoti i te x=x^2-30

$$x = x ^ { 2 } - 30$$

Whakaoti mō x

$$x=-5$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te x^{2} mai i ngā taha e rua.

$$x-x^{2}=-30$$

Me tāpiri te 30 ki ngā taha e rua.

$$x-x^{2}+30=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+x+30=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=-30=-30$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=6 b=-5$$

Tuhia anō te -x^{2}+x+30 hei \left(-x^{2}+6x\right)+\left(-5x+30\right).

$$\left(-x^{2}+6x\right)+\left(-5x+30\right)$$

Tauwehea te -x i te tuatahi me te -5 i te rōpū tuarua.

$$-x\left(x-6\right)-5\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(-x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te -x-5=0.

$$x=6 x=-5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te x^{2} mai i ngā taha e rua.

$$x-x^{2}=-30$$

Me tāpiri te 30 ki ngā taha e rua.

$$x-x^{2}+30=0$$

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.

$$-x^{2}+x+30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 1 mō b, me 30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-1±\sqrt{1+4\times 30}}{2\left(-1\right)}$$

Whakareatia 4 ki te 30.

$$x=\frac{-1±\sqrt{1+120}}{2\left(-1\right)}$$

Tāpiri 1 ki te 120.

$$x=\frac{-1±\sqrt{121}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-1±11}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-1±11}{-2}$$

Nā, me whakaoti te whārite x=\frac{-1±11}{-2} ina he tāpiri te ±. Tāpiri -1 ki te 11.

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

Whakawehe 10 ki te -2.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-1±11}{-2} ina he tango te ±. Tango 11 mai i -1.

$$x=-\frac{12}{-2}$$

Whakawehe -12 ki te -2.

$$x=6$$

Kua oti te whārite te whakatau.

$$x=-5 x=6$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te x^{2} mai i ngā taha e rua.

$$x-x^{2}=-30$$

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.

$$-x^{2}+x=-30$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+x}{-1}=-\frac{30}{-1}$$

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

$$x^{2}+\frac{1}{-1}x=-\frac{30}{-1}$$

Whakawehe 1 ki te -1.

$$x^{2}-x=-\frac{30}{-1}$$

Whakawehe -30 ki te -1.

$$x^{2}-x=30$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=30+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=30+\frac{1}{4}$$

Tāpiri 30 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{121}{4}$$

Tauwehea x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{11}{2} x-\frac{1}{2}=-\frac{11}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=6 x=-5$$