Whakaoti i te x^2-x=42

$$x ^ { 2 } - x = 42$$

Whakaoti mō x

$$x=-6$$

$$x=7$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 42 mai i ngā taha e rua.

$$x^{2}-x-42=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-x-42 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.

$$a+b=-1 ab=-42$$

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 -42.

$$1,-42 2,-21 3,-14 6,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-42=-41 2-21=-19 3-14=-11 6-7=-1$$

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

$$a=-7 b=6$$

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

$$\left(x-7\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+6=0.

$$x=7 x=-6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 42 mai i ngā taha e rua.

$$x^{2}-x-42=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-42. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-42\right)=-42$$

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 -42.

$$1,-42 2,-21 3,-14 6,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-42=-41 2-21=-19 3-14=-11 6-7=-1$$

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

$$a=-7 b=6$$

Tuhia anō te x^{2}-x-42 hei \left(x^{2}-7x\right)+\left(6x-42\right).

$$\left(x^{2}-7x\right)+\left(6x-42\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x-7\right)+6\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+6=0.

$$x=7 x=-6$$

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.

$$x^{2}-x=42$$

Me tango 42 mai i ngā taha e rua o te whārite.

$$x^{2}-x-42=42-42$$

Mā te tango i te 42 i a ia ake anō ka toe ko te 0.

$$x^{2}-x-42=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -42 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-42\right)}}{2}$$

Whakareatia -4 ki te -42.

$$x=\frac{-\left(-1\right)±\sqrt{1+168}}{2}$$

Tāpiri 1 ki te 168.

$$x=\frac{-\left(-1\right)±\sqrt{169}}{2}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-1\right)±13}{2}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±13}{2}$$

Nā, me whakaoti te whārite x=\frac{1±13}{2} ina he tāpiri te ±. Tāpiri 1 ki te 13.

$$x=\frac{14}{2}$$

Whakawehe 14 ki te 2.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{1±13}{2} ina he tango te ±. Tango 13 mai i 1.

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

Whakawehe -12 ki te 2.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=7 x=-6$$

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.

$$x^{2}-x=42$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=42+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=42+\frac{1}{4}$$

Tāpiri 42 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{169}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{169}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{13}{2} x-\frac{1}{2}=-\frac{13}{2}$$

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

$$x=7 x=-6$$