Whakaoti i te y^2-17y+30=0

$$y ^ { 2 } - 17 y + 30 = 0$$

Whakaoti mō y

$$y=2$$

$$y=15$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te y^{2}-17y+30 mā te whakamahi i te tātai y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-17 ab=30$$

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 30.

$$-1,-30 -2,-15 -3,-10 -5,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11$$

Ko te otinga te takirua ka hoatu i te tapeke -17.

$$a=-15 b=-2$$

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

$$\left(y-15\right)\left(y-2\right)$$

Hei kimi otinga whārite, me whakaoti te y-15=0 me te y-2=0.

$$y=15 y=2$$

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 y^{2}+ay+by+30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-17 ab=1\times 30=30$$

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 30.

$$-1,-30 -2,-15 -3,-10 -5,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11$$

Ko te otinga te takirua ka hoatu i te tapeke -17.

$$a=-15 b=-2$$

Tuhia anō te y^{2}-17y+30 hei \left(y^{2}-15y\right)+\left(-2y+30\right).

$$\left(y^{2}-15y\right)+\left(-2y+30\right)$$

Tauwehea te y i te tuatahi me te -2 i te rōpū tuarua.

$$y\left(y-15\right)-2\left(y-15\right)$$

Whakatauwehea atu te kīanga pātahi y-15 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-15\right)\left(y-2\right)$$

Hei kimi otinga whārite, me whakaoti te y-15=0 me te y-2=0.

$$y=15 y=2$$

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.

$$y^{2}-17y+30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -17 mō b, me 30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 30}}{2}$$

Pūrua -17.

$$y=\frac{-\left(-17\right)±\sqrt{289-4\times 30}}{2}$$

Whakareatia -4 ki te 30.

$$y=\frac{-\left(-17\right)±\sqrt{289-120}}{2}$$

Tāpiri 289 ki te -120.

$$y=\frac{-\left(-17\right)±\sqrt{169}}{2}$$

Tuhia te pūtakerua o te 169.

$$y=\frac{-\left(-17\right)±13}{2}$$

Ko te tauaro o -17 ko 17.

$$y=\frac{17±13}{2}$$

Nā, me whakaoti te whārite y=\frac{17±13}{2} ina he tāpiri te ±. Tāpiri 17 ki te 13.

$$y=\frac{30}{2}$$

Whakawehe 30 ki te 2.

$$y=15$$

Nā, me whakaoti te whārite y=\frac{17±13}{2} ina he tango te ±. Tango 13 mai i 17.

$$y=\frac{4}{2}$$

Whakawehe 4 ki te 2.

$$y=2$$

Kua oti te whārite te whakatau.

$$y=15 y=2$$

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.

$$y^{2}-17y+30=0$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$y^{2}-17y+30-30=-30$$

Mā te tango i te 30 i a ia ake anō ka toe ko te 0.

$$y^{2}-17y=-30$$

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

$$y^{2}-17y+\left(-\frac{17}{2}\right)^{2}=-30+\left(-\frac{17}{2}\right)^{2}$$

Pūruatia -\frac{17}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-17y+\frac{289}{4}=-30+\frac{289}{4}$$

Tāpiri -30 ki te \frac{289}{4}.

$$y^{2}-17y+\frac{289}{4}=\frac{169}{4}$$

Tauwehea y^{2}-17y+\frac{289}{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}.

$$\left(y-\frac{17}{2}\right)^{2}=\frac{169}{4}$$

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

$$\sqrt{\left(y-\frac{17}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

$$y-\frac{17}{2}=\frac{13}{2} y-\frac{17}{2}=-\frac{13}{2}$$

Me tāpiri \frac{17}{2} ki ngā taha e rua o te whārite.

$$y=15 y=2$$