Whakaoti i te (3+r)^2+(15+r)^2=18^2

$$( 3 + r ) ^ { 2 } + ( 15 + r ) ^ { 2 } = 18 ^ { 2 }$$

Whakaoti mō r

$$r=3\sqrt{14}-9\approx 2.22497216$$

$$r=-3\sqrt{14}-9\approx -20.22497216$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3+r\right)^{2}.

$$9+6r+r^{2}+\left(15+r\right)^{2}=18^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(15+r\right)^{2}.

$$9+6r+r^{2}+225+30r+r^{2}=18^{2}$$

Tāpirihia te 9 ki te 225, ka 234.

$$234+6r+r^{2}+30r+r^{2}=18^{2}$$

Pahekotia te 6r me 30r, ka 36r.

$$234+36r+r^{2}+r^{2}=18^{2}$$

Pahekotia te r^{2} me r^{2}, ka 2r^{2}.

$$234+36r+2r^{2}=18^{2}$$

Tātaihia te 18 mā te pū o 2, kia riro ko 324.

$$234+36r+2r^{2}=324$$

Tangohia te 324 mai i ngā taha e rua.

$$234+36r+2r^{2}-324=0$$

Tangohia te 324 i te 234, ka -90.

$$-90+36r+2r^{2}=0$$

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.

$$2r^{2}+36r-90=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 36 mō b, me -90 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-36±\sqrt{36^{2}-4\times 2\left(-90\right)}}{2\times 2}$$

Pūrua 36.

$$r=\frac{-36±\sqrt{1296-4\times 2\left(-90\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$r=\frac{-36±\sqrt{1296-8\left(-90\right)}}{2\times 2}$$

Whakareatia -8 ki te -90.

$$r=\frac{-36±\sqrt{1296+720}}{2\times 2}$$

Tāpiri 1296 ki te 720.

$$r=\frac{-36±\sqrt{2016}}{2\times 2}$$

Tuhia te pūtakerua o te 2016.

$$r=\frac{-36±12\sqrt{14}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$r=\frac{-36±12\sqrt{14}}{4}$$

Nā, me whakaoti te whārite r=\frac{-36±12\sqrt{14}}{4} ina he tāpiri te ±. Tāpiri -36 ki te 12\sqrt{14}.

$$r=\frac{12\sqrt{14}-36}{4}$$

Whakawehe -36+12\sqrt{14} ki te 4.

$$r=3\sqrt{14}-9$$

Nā, me whakaoti te whārite r=\frac{-36±12\sqrt{14}}{4} ina he tango te ±. Tango 12\sqrt{14} mai i -36.

$$r=\frac{-12\sqrt{14}-36}{4}$$

Whakawehe -36-12\sqrt{14} ki te 4.

$$r=-3\sqrt{14}-9$$

Kua oti te whārite te whakatau.

$$r=3\sqrt{14}-9 r=-3\sqrt{14}-9$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3+r\right)^{2}.

$$9+6r+r^{2}+\left(15+r\right)^{2}=18^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(15+r\right)^{2}.

$$9+6r+r^{2}+225+30r+r^{2}=18^{2}$$

Tāpirihia te 9 ki te 225, ka 234.

$$234+6r+r^{2}+30r+r^{2}=18^{2}$$

Pahekotia te 6r me 30r, ka 36r.

$$234+36r+r^{2}+r^{2}=18^{2}$$

Pahekotia te r^{2} me r^{2}, ka 2r^{2}.

$$234+36r+2r^{2}=18^{2}$$

Tātaihia te 18 mā te pū o 2, kia riro ko 324.

$$234+36r+2r^{2}=324$$

Tangohia te 234 mai i ngā taha e rua.

$$36r+2r^{2}=324-234$$

Tangohia te 234 i te 324, ka 90.

$$36r+2r^{2}=90$$

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.

$$2r^{2}+36r=90$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2r^{2}+36r}{2}=\frac{90}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$r^{2}+\frac{36}{2}r=\frac{90}{2}$$

Whakawehe 36 ki te 2.

$$r^{2}+18r=\frac{90}{2}$$

Whakawehe 90 ki te 2.

$$r^{2}+18r=45$$

Whakawehea te 18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 9. Nā, tāpiria te pūrua o te 9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}+18r+9^{2}=45+9^{2}$$

Pūrua 9.

$$r^{2}+18r+81=45+81$$

Tāpiri 45 ki te 81.

$$r^{2}+18r+81=126$$

Tauwehea r^{2}+18r+81. 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}.

$$\left(r+9\right)^{2}=126$$

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

$$\sqrt{\left(r+9\right)^{2}}=\sqrt{126}$$

Whakarūnātia.

$$r+9=3\sqrt{14} r+9=-3\sqrt{14}$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$r=3\sqrt{14}-9 r=-3\sqrt{14}-9$$