Whakaoti i te x^2-15x+44=0

$$x ^ { 2 } - 15 x + 44 = 0$$

Whakaoti mō x

$$x=4$$

$$x=11$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-15x+44 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.

$$a+b=-15 ab=44$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 44.

$$-1,-44 -2,-22 -4,-11$$

Tātaihia te tapeke mō ia takirua.

$$-1-44=-45 -2-22=-24 -4-11=-15$$

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

$$a=-11 b=-4$$

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

$$\left(x-11\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-11=0 me te x-4=0.

$$x=11 x=4$$

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+44. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-15 ab=1\times 44=44$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 44.

$$-1,-44 -2,-22 -4,-11$$

Tātaihia te tapeke mō ia takirua.

$$-1-44=-45 -2-22=-24 -4-11=-15$$

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

$$a=-11 b=-4$$

Tuhia anō te x^{2}-15x+44 hei \left(x^{2}-11x\right)+\left(-4x+44\right).

$$\left(x^{2}-11x\right)+\left(-4x+44\right)$$

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

$$x\left(x-11\right)-4\left(x-11\right)$$

Whakatauwehea atu te kīanga pātahi x-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-11\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-11=0 me te x-4=0.

$$x=11 x=4$$

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.

$$x^{2}-15x+44=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -15 mō b, me 44 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 44}}{2}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 44}}{2}$$

Whakareatia -4 ki te 44.

$$x=\frac{-\left(-15\right)±\sqrt{225-176}}{2}$$

Tāpiri 225 ki te -176.

$$x=\frac{-\left(-15\right)±\sqrt{49}}{2}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-\left(-15\right)±7}{2}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±7}{2}$$

Nā, me whakaoti te whārite x=\frac{15±7}{2} ina he tāpiri te ±. Tāpiri 15 ki te 7.

$$x=\frac{22}{2}$$

Whakawehe 22 ki te 2.

$$x=11$$

Nā, me whakaoti te whārite x=\frac{15±7}{2} ina he tango te ±. Tango 7 mai i 15.

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

Whakawehe 8 ki te 2.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=11 x=4$$

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.

$$x^{2}-15x+44=0$$

Me tango 44 mai i ngā taha e rua o te whārite.

$$x^{2}-15x+44-44=-44$$

Mā te tango i te 44 i a ia ake anō ka toe ko te 0.

$$x^{2}-15x=-44$$

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.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-44+\left(-\frac{15}{2}\right)^{2}$$

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

$$x^{2}-15x+\frac{225}{4}=-44+\frac{225}{4}$$

Tāpiri -44 ki te \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{49}{4}$$

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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x-\frac{15}{2}=\frac{7}{2} x-\frac{15}{2}=-\frac{7}{2}$$

Me tāpiri \frac{15}{2} ki ngā taha e rua o te whārite.

$$x=11 x=4$$