Whakaoti i te x^2+15x+54=0

$${ x }^{ 2 } +15x+54=0$$

Whakaoti mō x

$$x=-9$$

$$x=-6$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+15x+54 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.

$$a+b=15 ab=54$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 54.

$$1,54 2,27 3,18 6,9$$

Tātaihia te tapeke mō ia takirua.

$$1+54=55 2+27=29 3+18=21 6+9=15$$

Ko te otinga te takirua ka hoatu i te tapeke 15.

$$a=6 b=9$$

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

$$\left(x+6\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x+6=0 me te x+9=0.

$$x=-6 x=-9$$

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+54. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=15 ab=1\times 54=54$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 54.

$$1,54 2,27 3,18 6,9$$

Tātaihia te tapeke mō ia takirua.

$$1+54=55 2+27=29 3+18=21 6+9=15$$

Ko te otinga te takirua ka hoatu i te tapeke 15.

$$a=6 b=9$$

Tuhia anō te x^{2}+15x+54 hei \left(x^{2}+6x\right)+\left(9x+54\right).

$$\left(x^{2}+6x\right)+\left(9x+54\right)$$

Tauwehea te x i te tuatahi me te 9 i te rōpū tuarua.

$$x\left(x+6\right)+9\left(x+6\right)$$

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

$$\left(x+6\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x+6=0 me te x+9=0.

$$x=-6 x=-9$$

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.

$$x^{2}+15x+54=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 15 mō b, me 54 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 54}}{2}$$

Pūrua 15.

$$x=\frac{-15±\sqrt{225-4\times 54}}{2}$$

Whakareatia -4 ki te 54.

$$x=\frac{-15±\sqrt{225-216}}{2}$$

Tāpiri 225 ki te -216.

$$x=\frac{-15±\sqrt{9}}{2}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-15±3}{2}$$

Nā, me whakaoti te whārite x=\frac{-15±3}{2} ina he tāpiri te ±. Tāpiri -15 ki te 3.

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

Whakawehe -12 ki te 2.

$$x=-6$$

Nā, me whakaoti te whārite x=\frac{-15±3}{2} ina he tango te ±. Tango 3 mai i -15.

$$x=-\frac{18}{2}$$

Whakawehe -18 ki te 2.

$$x=-9$$

Kua oti te whārite te whakatau.

$$x=-6 x=-9$$

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.

$$x^{2}+15x+54=0$$

Me tango 54 mai i ngā taha e rua o te whārite.

$$x^{2}+15x+54-54=-54$$

Mā te tango i te 54 i a ia ake anō ka toe ko te 0.

$$x^{2}+15x=-54$$

Whakawehea te 15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{2}. Nā, tāpiria te pūrua o te \frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=-54+\left(\frac{15}{2}\right)^{2}$$

Pūruatia \frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+15x+\frac{225}{4}=-54+\frac{225}{4}$$

Tāpiri -54 ki te \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{9}{4}$$

Tauwehea x^{2}+15x+\frac{225}{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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

$$x+\frac{15}{2}=\frac{3}{2} x+\frac{15}{2}=-\frac{3}{2}$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$x=-6 x=-9$$