Whakaoti i te x^2+34x+240=0

$${ x }^{ 2 } +34x+240=0$$

Whakaoti mō x

$$x=-24$$

$$x=-10$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+34x+240 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.

$$a+b=34 ab=240$$

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 240.

$$1,240 2,120 3,80 4,60 5,48 6,40 8,30 10,24 12,20 15,16$$

Tātaihia te tapeke mō ia takirua.

$$1+240=241 2+120=122 3+80=83 4+60=64 5+48=53 6+40=46 8+30=38 10+24=34 12+20=32 15+16=31$$

Ko te otinga te takirua ka hoatu i te tapeke 34.

$$a=10 b=24$$

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

$$\left(x+10\right)\left(x+24\right)$$

Hei kimi otinga whārite, me whakaoti te x+10=0 me te x+24=0.

$$x=-10 x=-24$$

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+240. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=34 ab=1\times 240=240$$

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 240.

$$1,240 2,120 3,80 4,60 5,48 6,40 8,30 10,24 12,20 15,16$$

Tātaihia te tapeke mō ia takirua.

$$1+240=241 2+120=122 3+80=83 4+60=64 5+48=53 6+40=46 8+30=38 10+24=34 12+20=32 15+16=31$$

Ko te otinga te takirua ka hoatu i te tapeke 34.

$$a=10 b=24$$

Tuhia anō te x^{2}+34x+240 hei \left(x^{2}+10x\right)+\left(24x+240\right).

$$\left(x^{2}+10x\right)+\left(24x+240\right)$$

Tauwehea te x i te tuatahi me te 24 i te rōpū tuarua.

$$x\left(x+10\right)+24\left(x+10\right)$$

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

$$\left(x+10\right)\left(x+24\right)$$

Hei kimi otinga whārite, me whakaoti te x+10=0 me te x+24=0.

$$x=-10 x=-24$$

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.

$$x^{2}+34x+240=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 34 mō b, me 240 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\times 240}}{2}$$

Pūrua 34.

$$x=\frac{-34±\sqrt{1156-4\times 240}}{2}$$

Whakareatia -4 ki te 240.

$$x=\frac{-34±\sqrt{1156-960}}{2}$$

Tāpiri 1156 ki te -960.

$$x=\frac{-34±\sqrt{196}}{2}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-34±14}{2}$$

Nā, me whakaoti te whārite x=\frac{-34±14}{2} ina he tāpiri te ±. Tāpiri -34 ki te 14.

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

Whakawehe -20 ki te 2.

$$x=-10$$

Nā, me whakaoti te whārite x=\frac{-34±14}{2} ina he tango te ±. Tango 14 mai i -34.

$$x=-\frac{48}{2}$$

Whakawehe -48 ki te 2.

$$x=-24$$

Kua oti te whārite te whakatau.

$$x=-10 x=-24$$

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.

$$x^{2}+34x+240=0$$

Me tango 240 mai i ngā taha e rua o te whārite.

$$x^{2}+34x+240-240=-240$$

Mā te tango i te 240 i a ia ake anō ka toe ko te 0.

$$x^{2}+34x=-240$$

Whakawehea te 34, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 17. Nā, tāpiria te pūrua o te 17 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+34x+17^{2}=-240+17^{2}$$

Pūrua 17.

$$x^{2}+34x+289=-240+289$$

Tāpiri -240 ki te 289.

$$x^{2}+34x+289=49$$

Tauwehea x^{2}+34x+289. 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}.

$$\left(x+17\right)^{2}=49$$

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

$$\sqrt{\left(x+17\right)^{2}}=\sqrt{49}$$

Whakarūnātia.

$$x+17=7 x+17=-7$$

Me tango 17 mai i ngā taha e rua o te whārite.

$$x=-10 x=-24$$