Whakaoti i te x^2+7x=8

$${ x }^{ 2 } +7x=8$$

Whakaoti mō x

$$x=-8$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 8 mai i ngā taha e rua.

$$x^{2}+7x-8=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+7x-8 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.

$$a+b=7 ab=-8$$

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 -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

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

$$a=-1 b=8$$

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

$$\left(x-1\right)\left(x+8\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te x+8=0.

$$x=1 x=-8$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 8 mai i ngā taha e rua.

$$x^{2}+7x-8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=1\left(-8\right)=-8$$

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 -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

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

$$a=-1 b=8$$

Tuhia anō te x^{2}+7x-8 hei \left(x^{2}-x\right)+\left(8x-8\right).

$$\left(x^{2}-x\right)+\left(8x-8\right)$$

Tauwehea te x i te tuatahi me te 8 i te rōpū tuarua.

$$x\left(x-1\right)+8\left(x-1\right)$$

Whakatauwehea atu te kīanga pātahi x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-1\right)\left(x+8\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te x+8=0.

$$x=1 x=-8$$

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.

$$x^{2}+7x=8$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$x^{2}+7x-8=8-8$$

Mā te tango i te 8 i a ia ake anō ka toe ko te 0.

$$x^{2}+7x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 7 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-8\right)}}{2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-8\right)}}{2}$$

Whakareatia -4 ki te -8.

$$x=\frac{-7±\sqrt{49+32}}{2}$$

Tāpiri 49 ki te 32.

$$x=\frac{-7±\sqrt{81}}{2}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-7±9}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±9}{2} ina he tāpiri te ±. Tāpiri -7 ki te 9.

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

Whakawehe 2 ki te 2.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-7±9}{2} ina he tango te ±. Tango 9 mai i -7.

$$x=-\frac{16}{2}$$

Whakawehe -16 ki te 2.

$$x=-8$$

Kua oti te whārite te whakatau.

$$x=1 x=-8$$

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.

$$x^{2}+7x=8$$

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.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=8+\left(\frac{7}{2}\right)^{2}$$

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

$$x^{2}+7x+\frac{49}{4}=8+\frac{49}{4}$$

Tāpiri 8 ki te \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{81}{4}$$

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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{81}{4}$$

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

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Whakarūnātia.

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

Me tango \frac{7}{2} mai i ngā taha e rua o te whārite.

$$x=1 x=-8$$