Whakaoti i te 3r^2-5(r+1)=7r+58

$$3 r ^ { 2 } - 5 ( r + 1 ) = 7 r + 58$$

Whakaoti mō r

$$r=-3$$

$$r=7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te r+1.

$$3r^{2}-5r-5=7r+58$$

Tangohia te 7r mai i ngā taha e rua.

$$3r^{2}-5r-5-7r=58$$

Pahekotia te -5r me -7r, ka -12r.

$$3r^{2}-12r-5=58$$

Tangohia te 58 mai i ngā taha e rua.

$$3r^{2}-12r-5-58=0$$

Tangohia te 58 i te -5, ka -63.

$$3r^{2}-12r-63=0$$

Whakawehea ngā taha e rua ki te 3.

$$r^{2}-4r-21=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei r^{2}+ar+br-21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=1\left(-21\right)=-21$$

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 -21.

$$1,-21 3,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-21=-20 3-7=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-7 b=3$$

Tuhia anō te r^{2}-4r-21 hei \left(r^{2}-7r\right)+\left(3r-21\right).

$$\left(r^{2}-7r\right)+\left(3r-21\right)$$

Tauwehea te r i te tuatahi me te 3 i te rōpū tuarua.

$$r\left(r-7\right)+3\left(r-7\right)$$

Whakatauwehea atu te kīanga pātahi r-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(r-7\right)\left(r+3\right)$$

Hei kimi otinga whārite, me whakaoti te r-7=0 me te r+3=0.

$$r=7 r=-3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te r+1.

$$3r^{2}-5r-5=7r+58$$

Tangohia te 7r mai i ngā taha e rua.

$$3r^{2}-5r-5-7r=58$$

Pahekotia te -5r me -7r, ka -12r.

$$3r^{2}-12r-5=58$$

Tangohia te 58 mai i ngā taha e rua.

$$3r^{2}-12r-5-58=0$$

Tangohia te 58 i te -5, ka -63.

$$3r^{2}-12r-63=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -12 mō b, me -63 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 3\left(-63\right)}}{2\times 3}$$

Pūrua -12.

$$r=\frac{-\left(-12\right)±\sqrt{144-4\times 3\left(-63\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$r=\frac{-\left(-12\right)±\sqrt{144-12\left(-63\right)}}{2\times 3}$$

Whakareatia -12 ki te -63.

$$r=\frac{-\left(-12\right)±\sqrt{144+756}}{2\times 3}$$

Tāpiri 144 ki te 756.

$$r=\frac{-\left(-12\right)±\sqrt{900}}{2\times 3}$$

Tuhia te pūtakerua o te 900.

$$r=\frac{-\left(-12\right)±30}{2\times 3}$$

Ko te tauaro o -12 ko 12.

$$r=\frac{12±30}{2\times 3}$$

Whakareatia 2 ki te 3.

$$r=\frac{12±30}{6}$$

Nā, me whakaoti te whārite r=\frac{12±30}{6} ina he tāpiri te ±. Tāpiri 12 ki te 30.

$$r=\frac{42}{6}$$

Whakawehe 42 ki te 6.

$$r=7$$

Nā, me whakaoti te whārite r=\frac{12±30}{6} ina he tango te ±. Tango 30 mai i 12.

$$r=-\frac{18}{6}$$

Whakawehe -18 ki te 6.

$$r=-3$$

Kua oti te whārite te whakatau.

$$r=7 r=-3$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te r+1.

$$3r^{2}-5r-5=7r+58$$

Tangohia te 7r mai i ngā taha e rua.

$$3r^{2}-5r-5-7r=58$$

Pahekotia te -5r me -7r, ka -12r.

$$3r^{2}-12r-5=58$$

Me tāpiri te 5 ki ngā taha e rua.

$$3r^{2}-12r=58+5$$

Tāpirihia te 58 ki te 5, ka 63.

$$3r^{2}-12r=63$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3r^{2}-12r}{3}=\frac{63}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$r^{2}+\left(-\frac{12}{3}\right)r=\frac{63}{3}$$

Whakawehe -12 ki te 3.

$$r^{2}-4r=\frac{63}{3}$$

Whakawehe 63 ki te 3.

$$r^{2}-4r=21$$

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}-4r+\left(-2\right)^{2}=21+\left(-2\right)^{2}$$

Pūrua -2.

$$r^{2}-4r+4=21+4$$

Tāpiri 21 ki te 4.

$$r^{2}-4r+4=25$$

Tauwehea r^{2}-4r+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}.

$$\left(r-2\right)^{2}=25$$

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

$$\sqrt{\left(r-2\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$r-2=5 r-2=-5$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$r=7 r=-3$$