Whakaoti i te y^2-36=5y

$$y ^ { 2 } - 36 = 5 y$$

Whakaoti mō y

$$y=-4$$

$$y=9$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 5y mai i ngā taha e rua.

$$y^{2}-36-5y=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$y^{2}-5y-36=0$$

Hei whakaoti i te whārite, whakatauwehea te y^{2}-5y-36 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.

$$a+b=-5 ab=-36$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-9 b=4$$

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

$$\left(y-9\right)\left(y+4\right)$$

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

$$y=9 y=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 5y mai i ngā taha e rua.

$$y^{2}-36-5y=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$y^{2}-5y-36=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei y^{2}+ay+by-36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=1\left(-36\right)=-36$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-9 b=4$$

Tuhia anō te y^{2}-5y-36 hei \left(y^{2}-9y\right)+\left(4y-36\right).

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

Tauwehea te y i te tuatahi me te 4 i te rōpū tuarua.

$$y\left(y-9\right)+4\left(y-9\right)$$

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

$$\left(y-9\right)\left(y+4\right)$$

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

$$y=9 y=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 5y mai i ngā taha e rua.

$$y^{2}-36-5y=0$$

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.

$$y^{2}-5y-36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -5 mō b, me -36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-36\right)}}{2}$$

Pūrua -5.

$$y=\frac{-\left(-5\right)±\sqrt{25-4\left(-36\right)}}{2}$$

Whakareatia -4 ki te -36.

$$y=\frac{-\left(-5\right)±\sqrt{25+144}}{2}$$

Tāpiri 25 ki te 144.

$$y=\frac{-\left(-5\right)±\sqrt{169}}{2}$$

Tuhia te pūtakerua o te 169.

$$y=\frac{-\left(-5\right)±13}{2}$$

Ko te tauaro o -5 ko 5.

$$y=\frac{5±13}{2}$$

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

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

Whakawehe 18 ki te 2.

$$y=9$$

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

$$y=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$y=-4$$

Kua oti te whārite te whakatau.

$$y=9 y=-4$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 5y mai i ngā taha e rua.

$$y^{2}-36-5y=0$$

Me tāpiri te 36 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y^{2}-5y=36$$

Whakawehea te -5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{2}. Nā, tāpiria te pūrua o te -\frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-5y+\left(-\frac{5}{2}\right)^{2}=36+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-5y+\frac{25}{4}=36+\frac{25}{4}$$

Tāpiri 36 ki te \frac{25}{4}.

$$y^{2}-5y+\frac{25}{4}=\frac{169}{4}$$

Tauwehea y^{2}-5y+\frac{25}{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}.

$$\left(y-\frac{5}{2}\right)^{2}=\frac{169}{4}$$

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

$$\sqrt{\left(y-\frac{5}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

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

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$y=9 y=-4$$