Whakaoti i te x^2-x/90=2/15

$$\frac { x ^ { 2 } - x } { 90 } = \frac { 2 } { 15 }$$

Whakaoti mō x

$$x=-3$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe

Me whakarea ngā taha e rua ki te 90.

$$x^{2}-x=\frac{2}{15}\times 90$$

Whakareatia te \frac{2}{15} ki te 90, ka 12.

$$x^{2}-x=12$$

Tangohia te 12 mai i ngā taha e rua.

$$x^{2}-x-12=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-x-12 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

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

$$a=-4 b=3$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-4\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x+3=0.

$$x=4 x=-3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me whakarea ngā taha e rua ki te 90.

$$x^{2}-x=\frac{2}{15}\times 90$$

Whakareatia te \frac{2}{15} ki te 90, ka 12.

$$x^{2}-x=12$$

Tangohia te 12 mai i ngā taha e rua.

$$x^{2}-x-12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-12\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

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

$$a=-4 b=3$$

Tuhia anō te x^{2}-x-12 hei \left(x^{2}-4x\right)+\left(3x-12\right).

$$\left(x^{2}-4x\right)+\left(3x-12\right)$$

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

$$x\left(x-4\right)+3\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x+3=0.

$$x=4 x=-3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua ki te 90.

$$x^{2}-x=\frac{2}{15}\times 90$$

Whakareatia te \frac{2}{15} ki te 90, ka 12.

$$x^{2}-x=12$$

Tangohia te 12 mai i ngā taha e rua.

$$x^{2}-x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-12\right)}}{2}$$

Whakareatia -4 ki te -12.

$$x=\frac{-\left(-1\right)±\sqrt{1+48}}{2}$$

Tāpiri 1 ki te 48.

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

Tuhia te pūtakerua o te 49.

$$x=\frac{-\left(-1\right)±7}{2}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±7}{2}$$

Nā, me whakaoti te whārite x=\frac{1±7}{2} ina he tāpiri te ±. Tāpiri 1 ki te 7.

$$x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{1±7}{2} ina he tango te ±. Tango 7 mai i 1.

$$x=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=4 x=-3$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua ki te 90.

$$x^{2}-x=\frac{2}{15}\times 90$$

Whakareatia te \frac{2}{15} ki te 90, ka 12.

$$x^{2}-x=12$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=12+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=12+\frac{1}{4}$$

Tāpiri 12 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{49}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{49}{4}$$

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

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

Whakarūnātia.

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

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

$$x=4 x=-3$$