Whakaoti i te 3x^2-15x=18

$$3 { x }^{ 2 } -15x=18$$

Whakaoti mō x

$$x=-1$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 18 mai i ngā taha e rua.

$$3x^{2}-15x-18=0$$

Whakawehea ngā taha e rua ki te 3.

$$x^{2}-5x-6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=1\left(-6\right)=-6$$

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 -6.

$$1,-6 2,-3$$

Tātaihia te tapeke mō ia takirua.

$$1-6=-5 2-3=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-6 b=1$$

Tuhia anō te x^{2}-5x-6 hei \left(x^{2}-6x\right)+\left(x-6\right).

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

Whakatauwehea atu x i te x^{2}-6x.

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

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x+1=0.

$$x=6 x=-1$$

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.

$$3x^{2}-15x=18$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$3x^{2}-15x-18=18-18$$

Mā te tango i te 18 i a ia ake anō ka toe ko te 0.

$$3x^{2}-15x-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -15 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 3\left(-18\right)}}{2\times 3}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 3\left(-18\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-15\right)±\sqrt{225-12\left(-18\right)}}{2\times 3}$$

Whakareatia -12 ki te -18.

$$x=\frac{-\left(-15\right)±\sqrt{225+216}}{2\times 3}$$

Tāpiri 225 ki te 216.

$$x=\frac{-\left(-15\right)±\sqrt{441}}{2\times 3}$$

Tuhia te pūtakerua o te 441.

$$x=\frac{-\left(-15\right)±21}{2\times 3}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±21}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{15±21}{6}$$

Nā, me whakaoti te whārite x=\frac{15±21}{6} ina he tāpiri te ±. Tāpiri 15 ki te 21.

$$x=\frac{36}{6}$$

Whakawehe 36 ki te 6.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{15±21}{6} ina he tango te ±. Tango 21 mai i 15.

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

Whakawehe -6 ki te 6.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=6 x=-1$$

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.

$$3x^{2}-15x=18$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-15x}{3}=\frac{18}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\left(-\frac{15}{3}\right)x=\frac{18}{3}$$

Whakawehe -15 ki te 3.

$$x^{2}-5x=\frac{18}{3}$$

Whakawehe 18 ki te 3.

$$x^{2}-5x=6$$

Whakawehea te -5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{2}. Nā, tāpiria te pūrua o te -\frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=6+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=6+\frac{25}{4}$$

Tāpiri 6 ki te \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{49}{4}$$

Tauwehea x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{7}{2} x-\frac{5}{2}=-\frac{7}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=6 x=-1$$