Whakaoti i te r^2-r=36+4r

$$r ^ { 2 } - r = 36 + 4 r$$

Whakaoti mō r

$$r=-4$$

$$r=9$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 36 mai i ngā taha e rua.

$$r^{2}-r-36=4r$$

Tangohia te 4r mai i ngā taha e rua.

$$r^{2}-r-36-4r=0$$

Pahekotia te -r me -4r, ka -5r.

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

Hei whakaoti i te whārite, whakatauwehea te r^{2}-5r-36 mā te whakamahi i te tātai r^{2}+\left(a+b\right)r+ab=\left(r+a\right)\left(r+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(r+a\right)\left(r+b\right) mā ngā uara i tātaihia.

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

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

$$r=9 r=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 36 mai i ngā taha e rua.

$$r^{2}-r-36=4r$$

Tangohia te 4r mai i ngā taha e rua.

$$r^{2}-r-36-4r=0$$

Pahekotia te -r me -4r, ka -5r.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei r^{2}+ar+br-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 r^{2}-5r-36 hei \left(r^{2}-9r\right)+\left(4r-36\right).

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

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

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

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

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

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

$$r=9 r=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 36 mai i ngā taha e rua.

$$r^{2}-r-36=4r$$

Tangohia te 4r mai i ngā taha e rua.

$$r^{2}-r-36-4r=0$$

Pahekotia te -r me -4r, ka -5r.

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

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}.

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

Pūrua -5.

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

Whakareatia -4 ki te -36.

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

Tāpiri 25 ki te 144.

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

Tuhia te pūtakerua o te 169.

$$r=\frac{-\left(-5\right)±13}{2}$$

Ko te tauaro o -5 ko 5.

$$r=\frac{5±13}{2}$$

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

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

Whakawehe 18 ki te 2.

$$r=9$$

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

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

Whakawehe -8 ki te 2.

$$r=-4$$

Kua oti te whārite te whakatau.

$$r=9 r=-4$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 4r mai i ngā taha e rua.

$$r^{2}-r-4r=36$$

Pahekotia te -r me -4r, ka -5r.

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

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.

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

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

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

Tāpiri 36 ki te \frac{25}{4}.

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

Tauwehea r^{2}-5r+\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}.

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

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

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

Whakarūnātia.

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

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

$$r=9 r=-4$$