Whakaoti i te (23-x)^2+x^2=17^2

$$( 23 - x ) ^ { 2 } + x ^ { 2 } = 17 ^ { 2 }$$

Whakaoti mō x

$$x=8$$

$$x=15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(23-x\right)^{2}.

$$529-46x+x^{2}+x^{2}=17^{2}$$

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

$$529-46x+2x^{2}=17^{2}$$

Tātaihia te 17 mā te pū o 2, kia riro ko 289.

$$529-46x+2x^{2}=289$$

Tangohia te 289 mai i ngā taha e rua.

$$529-46x+2x^{2}-289=0$$

Tangohia te 289 i te 529, ka 240.

$$240-46x+2x^{2}=0$$

Whakawehea ngā taha e rua ki te 2.

$$120-23x+x^{2}=0$$

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

$$x^{2}-23x+120=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+120. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-23 ab=1\times 120=120$$

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 120.

$$-1,-120 -2,-60 -3,-40 -4,-30 -5,-24 -6,-20 -8,-15 -10,-12$$

Tātaihia te tapeke mō ia takirua.

$$-1-120=-121 -2-60=-62 -3-40=-43 -4-30=-34 -5-24=-29 -6-20=-26 -8-15=-23 -10-12=-22$$

Ko te otinga te takirua ka hoatu i te tapeke -23.

$$a=-15 b=-8$$

Tuhia anō te x^{2}-23x+120 hei \left(x^{2}-15x\right)+\left(-8x+120\right).

$$\left(x^{2}-15x\right)+\left(-8x+120\right)$$

Tauwehea te x i te tuatahi me te -8 i te rōpū tuarua.

$$x\left(x-15\right)-8\left(x-15\right)$$

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

$$\left(x-15\right)\left(x-8\right)$$

Hei kimi otinga whārite, me whakaoti te x-15=0 me te x-8=0.

$$x=15 x=8$$

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(23-x\right)^{2}.

$$529-46x+x^{2}+x^{2}=17^{2}$$

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

$$529-46x+2x^{2}=17^{2}$$

Tātaihia te 17 mā te pū o 2, kia riro ko 289.

$$529-46x+2x^{2}=289$$

Tangohia te 289 mai i ngā taha e rua.

$$529-46x+2x^{2}-289=0$$

Tangohia te 289 i te 529, ka 240.

$$240-46x+2x^{2}=0$$

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.

$$2x^{2}-46x+240=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -46 mō b, me 240 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-46\right)±\sqrt{\left(-46\right)^{2}-4\times 2\times 240}}{2\times 2}$$

Pūrua -46.

$$x=\frac{-\left(-46\right)±\sqrt{2116-4\times 2\times 240}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-46\right)±\sqrt{2116-8\times 240}}{2\times 2}$$

Whakareatia -8 ki te 240.

$$x=\frac{-\left(-46\right)±\sqrt{2116-1920}}{2\times 2}$$

Tāpiri 2116 ki te -1920.

$$x=\frac{-\left(-46\right)±\sqrt{196}}{2\times 2}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-\left(-46\right)±14}{2\times 2}$$

Ko te tauaro o -46 ko 46.

$$x=\frac{46±14}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{46±14}{4}$$

Nā, me whakaoti te whārite x=\frac{46±14}{4} ina he tāpiri te ±. Tāpiri 46 ki te 14.

$$x=\frac{60}{4}$$

Whakawehe 60 ki te 4.

$$x=15$$

Nā, me whakaoti te whārite x=\frac{46±14}{4} ina he tango te ±. Tango 14 mai i 46.

$$x=\frac{32}{4}$$

Whakawehe 32 ki te 4.

$$x=8$$

Kua oti te whārite te whakatau.

$$x=15 x=8$$

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(23-x\right)^{2}.

$$529-46x+x^{2}+x^{2}=17^{2}$$

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

$$529-46x+2x^{2}=17^{2}$$

Tātaihia te 17 mā te pū o 2, kia riro ko 289.

$$529-46x+2x^{2}=289$$

Tangohia te 529 mai i ngā taha e rua.

$$-46x+2x^{2}=289-529$$

Tangohia te 529 i te 289, ka -240.

$$-46x+2x^{2}=-240$$

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.

$$2x^{2}-46x=-240$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-46x}{2}=-\frac{240}{2}$$

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

$$x^{2}+\left(-\frac{46}{2}\right)x=-\frac{240}{2}$$

Whakawehe -46 ki te 2.

$$x^{2}-23x=-\frac{240}{2}$$

Whakawehe -240 ki te 2.

$$x^{2}-23x=-120$$

Whakawehea te -23, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{23}{2}. Nā, tāpiria te pūrua o te -\frac{23}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-23x+\left(-\frac{23}{2}\right)^{2}=-120+\left(-\frac{23}{2}\right)^{2}$$

Pūruatia -\frac{23}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-23x+\frac{529}{4}=-120+\frac{529}{4}$$

Tāpiri -120 ki te \frac{529}{4}.

$$x^{2}-23x+\frac{529}{4}=\frac{49}{4}$$

Tauwehea x^{2}-23x+\frac{529}{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}.

$$\left(x-\frac{23}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{23}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x-\frac{23}{2}=\frac{7}{2} x-\frac{23}{2}=-\frac{7}{2}$$

Me tāpiri \frac{23}{2} ki ngā taha e rua o te whārite.

$$x=15 x=8$$