Whakaoti i te x^2-4x-60=0

$$x ^ { 2 } - 4 x - 60 = 0$$

Whakaoti mō x

$$x=-6$$

$$x=10$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-4x-60 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.

$$a+b=-4 ab=-60$$

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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-10 b=6$$

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

$$\left(x-10\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x+6=0.

$$x=10 x=-6$$

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-60. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=1\left(-60\right)=-60$$

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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-10 b=6$$

Tuhia anō te x^{2}-4x-60 hei \left(x^{2}-10x\right)+\left(6x-60\right).

$$\left(x^{2}-10x\right)+\left(6x-60\right)$$

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

$$x\left(x-10\right)+6\left(x-10\right)$$

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

$$\left(x-10\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x+6=0.

$$x=10 x=-6$$

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.

$$x^{2}-4x-60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -4 mō b, me -60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-60\right)}}{2}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-60\right)}}{2}$$

Whakareatia -4 ki te -60.

$$x=\frac{-\left(-4\right)±\sqrt{16+240}}{2}$$

Tāpiri 16 ki te 240.

$$x=\frac{-\left(-4\right)±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-4\right)±16}{2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±16}{2}$$

Nā, me whakaoti te whārite x=\frac{4±16}{2} ina he tāpiri te ±. Tāpiri 4 ki te 16.

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

Whakawehe 20 ki te 2.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{4±16}{2} ina he tango te ±. Tango 16 mai i 4.

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

Whakawehe -12 ki te 2.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=10 x=-6$$

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.

$$x^{2}-4x-60=0$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$x^{2}-4x-60-\left(-60\right)=-\left(-60\right)$$

Mā te tango i te -60 i a ia ake anō ka toe ko te 0.

$$x^{2}-4x=-\left(-60\right)$$

Tango -60 mai i 0.

$$x^{2}-4x=60$$

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-4x+\left(-2\right)^{2}=60+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=60+4$$

Tāpiri 60 ki te 4.

$$x^{2}-4x+4=64$$

Tauwehea x^{2}-4x+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}.

$$\left(x-2\right)^{2}=64$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$x-2=8 x-2=-8$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=10 x=-6$$