Whakaoti i te x^2+14x+45=0

$$x ^ { 2 } + 14 x + 45 = 0$$

Whakaoti mō x

$$x=-9$$

$$x=-5$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+14x+45 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.

$$a+b=14 ab=45$$

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 45.

$$1,45 3,15 5,9$$

Tātaihia te tapeke mō ia takirua.

$$1+45=46 3+15=18 5+9=14$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=5 b=9$$

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

$$\left(x+5\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x+5=0 me te x+9=0.

$$x=-5 x=-9$$

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+45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=1\times 45=45$$

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 45.

$$1,45 3,15 5,9$$

Tātaihia te tapeke mō ia takirua.

$$1+45=46 3+15=18 5+9=14$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=5 b=9$$

Tuhia anō te x^{2}+14x+45 hei \left(x^{2}+5x\right)+\left(9x+45\right).

$$\left(x^{2}+5x\right)+\left(9x+45\right)$$

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

$$x\left(x+5\right)+9\left(x+5\right)$$

Whakatauwehea atu te kīanga pātahi x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+5\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x+5=0 me te x+9=0.

$$x=-5 x=-9$$

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.

$$x^{2}+14x+45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 14 mō b, me 45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 45}}{2}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 45}}{2}$$

Whakareatia -4 ki te 45.

$$x=\frac{-14±\sqrt{196-180}}{2}$$

Tāpiri 196 ki te -180.

$$x=\frac{-14±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-14±4}{2}$$

Nā, me whakaoti te whārite x=\frac{-14±4}{2} ina he tāpiri te ±. Tāpiri -14 ki te 4.

$$x=-\frac{10}{2}$$

Whakawehe -10 ki te 2.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-14±4}{2} ina he tango te ±. Tango 4 mai i -14.

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

Whakawehe -18 ki te 2.

$$x=-9$$

Kua oti te whārite te whakatau.

$$x=-5 x=-9$$

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.

$$x^{2}+14x+45=0$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$x^{2}+14x+45-45=-45$$

Mā te tango i te 45 i a ia ake anō ka toe ko te 0.

$$x^{2}+14x=-45$$

Whakawehea te 14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 7. Nā, tāpiria te pūrua o te 7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+14x+7^{2}=-45+7^{2}$$

Pūrua 7.

$$x^{2}+14x+49=-45+49$$

Tāpiri -45 ki te 49.

$$x^{2}+14x+49=4$$

Tauwehea x^{2}+14x+49. 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}.

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

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

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x+7=2 x+7=-2$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x=-5 x=-9$$