Whakaoti i te s^2-13s+36=0

$$s ^ { 2 } - 13 s + 36 = 0$$

Whakaoti mō s

$$s=4$$

$$s=9$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te s^{2}-13s+36 mā te whakamahi i te tātai s^{2}+\left(a+b\right)s+ab=\left(s+a\right)\left(s+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-13 ab=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -13.

$$a=-9 b=-4$$

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

$$\left(s-9\right)\left(s-4\right)$$

Hei kimi otinga whārite, me whakaoti te s-9=0 me te s-4=0.

$$s=9 s=4$$

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 s^{2}+as+bs+36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-13 ab=1\times 36=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -13.

$$a=-9 b=-4$$

Tuhia anō te s^{2}-13s+36 hei \left(s^{2}-9s\right)+\left(-4s+36\right).

$$\left(s^{2}-9s\right)+\left(-4s+36\right)$$

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

$$s\left(s-9\right)-4\left(s-9\right)$$

Whakatauwehea atu te kīanga pātahi s-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(s-9\right)\left(s-4\right)$$

Hei kimi otinga whārite, me whakaoti te s-9=0 me te s-4=0.

$$s=9 s=4$$

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.

$$s^{2}-13s+36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -13 mō b, me 36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 36}}{2}$$

Pūrua -13.

$$s=\frac{-\left(-13\right)±\sqrt{169-4\times 36}}{2}$$

Whakareatia -4 ki te 36.

$$s=\frac{-\left(-13\right)±\sqrt{169-144}}{2}$$

Tāpiri 169 ki te -144.

$$s=\frac{-\left(-13\right)±\sqrt{25}}{2}$$

Tuhia te pūtakerua o te 25.

$$s=\frac{-\left(-13\right)±5}{2}$$

Ko te tauaro o -13 ko 13.

$$s=\frac{13±5}{2}$$

Nā, me whakaoti te whārite s=\frac{13±5}{2} ina he tāpiri te ±. Tāpiri 13 ki te 5.

$$s=\frac{18}{2}$$

Whakawehe 18 ki te 2.

$$s=9$$

Nā, me whakaoti te whārite s=\frac{13±5}{2} ina he tango te ±. Tango 5 mai i 13.

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

Whakawehe 8 ki te 2.

$$s=4$$

Kua oti te whārite te whakatau.

$$s=9 s=4$$

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.

$$s^{2}-13s+36=0$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$s^{2}-13s+36-36=-36$$

Mā te tango i te 36 i a ia ake anō ka toe ko te 0.

$$s^{2}-13s=-36$$

Whakawehea te -13, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{2}. Nā, tāpiria te pūrua o te -\frac{13}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$s^{2}-13s+\left(-\frac{13}{2}\right)^{2}=-36+\left(-\frac{13}{2}\right)^{2}$$

Pūruatia -\frac{13}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$s^{2}-13s+\frac{169}{4}=-36+\frac{169}{4}$$

Tāpiri -36 ki te \frac{169}{4}.

$$s^{2}-13s+\frac{169}{4}=\frac{25}{4}$$

Tauwehea s^{2}-13s+\frac{169}{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}.

$$\left(s-\frac{13}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(s-\frac{13}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$s-\frac{13}{2}=\frac{5}{2} s-\frac{13}{2}=-\frac{5}{2}$$

Me tāpiri \frac{13}{2} ki ngā taha e rua o te whārite.

$$s=9 s=4$$