Whakaoti i te x^2-7x-30=0

$$x ^ { 2 } - 7 x - 30 = 0$$

Whakaoti mō x

$$x=-3$$

$$x=10$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-7x-30 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.

$$a+b=-7 ab=-30$$

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 -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 -7.

$$a=-10 b=3$$

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

$$\left(x-10\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x+3=0.

$$x=10 x=-3$$

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-30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=1\left(-30\right)=-30$$

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 -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 -7.

$$a=-10 b=3$$

Tuhia anō te x^{2}-7x-30 hei \left(x^{2}-10x\right)+\left(3x-30\right).

$$\left(x^{2}-10x\right)+\left(3x-30\right)$$

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

$$x\left(x-10\right)+3\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x+3=0.

$$x=10 x=-3$$

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.

$$x^{2}-7x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -7 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-30\right)}}{2}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-30\right)}}{2}$$

Whakareatia -4 ki te -30.

$$x=\frac{-\left(-7\right)±\sqrt{49+120}}{2}$$

Tāpiri 49 ki te 120.

$$x=\frac{-\left(-7\right)±\sqrt{169}}{2}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-7\right)±13}{2}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±13}{2}$$

Nā, me whakaoti te whārite x=\frac{7±13}{2} ina he tāpiri te ±. Tāpiri 7 ki te 13.

$$x=\frac{20}{2}$$

Whakawehe 20 ki te 2.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{7±13}{2} ina he tango te ±. Tango 13 mai i 7.

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

Whakawehe -6 ki te 2.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=10 x=-3$$

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.

$$x^{2}-7x-30=0$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$x^{2}-7x-30-\left(-30\right)=-\left(-30\right)$$

Mā te tango i te -30 i a ia ake anō ka toe ko te 0.

$$x^{2}-7x=-\left(-30\right)$$

Tango -30 mai i 0.

$$x^{2}-7x=30$$

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.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=30+\left(-\frac{7}{2}\right)^{2}$$

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

$$x^{2}-7x+\frac{49}{4}=30+\frac{49}{4}$$

Tāpiri 30 ki te \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{169}{4}$$

Tauwehea te x^{2}-7x+\frac{49}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{169}{4}$$

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

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

$$x-\frac{7}{2}=\frac{13}{2} x-\frac{7}{2}=-\frac{13}{2}$$

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

$$x=10 x=-3$$