Whakaoti i te x^2-3x=28

$$x^2-3x=28$$

Whakaoti mō x

$$x=-4$$

$$x=7$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 28 mai i ngā taha e rua.

$$x^{2}-3x-28=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-3x-28 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.

$$a+b=-3 ab=-28$$

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 -28.

$$1,-28 2,-14 4,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-28=-27 2-14=-12 4-7=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-7 b=4$$

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

$$\left(x-7\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+4=0.

$$x=7 x=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 28 mai i ngā taha e rua.

$$x^{2}-3x-28=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=1\left(-28\right)=-28$$

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 -28.

$$1,-28 2,-14 4,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-28=-27 2-14=-12 4-7=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-7 b=4$$

Tuhia anō te x^{2}-3x-28 hei \left(x^{2}-7x\right)+\left(4x-28\right).

$$\left(x^{2}-7x\right)+\left(4x-28\right)$$

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(x-7\right)+4\left(x-7\right)$$

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

$$\left(x-7\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+4=0.

$$x=7 x=-4$$

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.

$$x^{2}-3x=28$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$x^{2}-3x-28=28-28$$

Mā te tango i te 28 i a ia ake anō ka toe ko te 0.

$$x^{2}-3x-28=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -3 mō b, me -28 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-28\right)}}{2}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-28\right)}}{2}$$

Whakareatia -4 ki te -28.

$$x=\frac{-\left(-3\right)±\sqrt{9+112}}{2}$$

Tāpiri 9 ki te 112.

$$x=\frac{-\left(-3\right)±\sqrt{121}}{2}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-\left(-3\right)±11}{2}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±11}{2}$$

Nā, me whakaoti te whārite x=\frac{3±11}{2} ina he tāpiri te ±. Tāpiri 3 ki te 11.

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

Whakawehe 14 ki te 2.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{3±11}{2} ina he tango te ±. Tango 11 mai i 3.

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

Whakawehe -8 ki te 2.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=7 x=-4$$

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.

$$x^{2}-3x=28$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=28+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=28+\frac{9}{4}$$

Tāpiri 28 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{121}{4}$$

Tauwehea x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{11}{2} x-\frac{3}{2}=-\frac{11}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=7 x=-4$$