Whakaoti i te 26^2=x^2+(x+14)^2

$$26 ^ { 2 } = x ^ { 2 } + ( x + 14 ) ^ { 2 }$$

Whakaoti mō x

$$x=-24$$

$$x=10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tātaihia te 26 mā te pū o 2, kia riro ko 676.

$$676=x^{2}+\left(x+14\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+14\right)^{2}.

$$676=x^{2}+x^{2}+28x+196$$

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

$$676=2x^{2}+28x+196$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2x^{2}+28x+196=676$$

Tangohia te 676 mai i ngā taha e rua.

$$2x^{2}+28x+196-676=0$$

Tangohia te 676 i te 196, ka -480.

$$2x^{2}+28x-480=0$$

Whakawehea ngā taha e rua ki te 2.

$$x^{2}+14x-240=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-240. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=1\left(-240\right)=-240$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -240.

$$-1,240 -2,120 -3,80 -4,60 -5,48 -6,40 -8,30 -10,24 -12,20 -15,16$$

Tātaihia te tapeke mō ia takirua.

$$-1+240=239 -2+120=118 -3+80=77 -4+60=56 -5+48=43 -6+40=34 -8+30=22 -10+24=14 -12+20=8 -15+16=1$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=-10 b=24$$

Tuhia anō te x^{2}+14x-240 hei \left(x^{2}-10x\right)+\left(24x-240\right).

$$\left(x^{2}-10x\right)+\left(24x-240\right)$$

Tauwehea te x i te tuatahi me te 24 i te rōpū tuarua.

$$x\left(x-10\right)+24\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(x+24\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x+24=0.

$$x=10 x=-24$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 26 mā te pū o 2, kia riro ko 676.

$$676=x^{2}+\left(x+14\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+14\right)^{2}.

$$676=x^{2}+x^{2}+28x+196$$

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

$$676=2x^{2}+28x+196$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2x^{2}+28x+196=676$$

Tangohia te 676 mai i ngā taha e rua.

$$2x^{2}+28x+196-676=0$$

Tangohia te 676 i te 196, ka -480.

$$2x^{2}+28x-480=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 28 mō b, me -480 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 2\left(-480\right)}}{2\times 2}$$

Pūrua 28.

$$x=\frac{-28±\sqrt{784-4\times 2\left(-480\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-28±\sqrt{784-8\left(-480\right)}}{2\times 2}$$

Whakareatia -8 ki te -480.

$$x=\frac{-28±\sqrt{784+3840}}{2\times 2}$$

Tāpiri 784 ki te 3840.

$$x=\frac{-28±\sqrt{4624}}{2\times 2}$$

Tuhia te pūtakerua o te 4624.

$$x=\frac{-28±68}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-28±68}{4}$$

Nā, me whakaoti te whārite x=\frac{-28±68}{4} ina he tāpiri te ±. Tāpiri -28 ki te 68.

$$x=\frac{40}{4}$$

Whakawehe 40 ki te 4.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{-28±68}{4} ina he tango te ±. Tango 68 mai i -28.

$$x=-\frac{96}{4}$$

Whakawehe -96 ki te 4.

$$x=-24$$

Kua oti te whārite te whakatau.

$$x=10 x=-24$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te 26 mā te pū o 2, kia riro ko 676.

$$676=x^{2}+\left(x+14\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+14\right)^{2}.

$$676=x^{2}+x^{2}+28x+196$$

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

$$676=2x^{2}+28x+196$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2x^{2}+28x+196=676$$

Tangohia te 196 mai i ngā taha e rua.

$$2x^{2}+28x=676-196$$

Tangohia te 196 i te 676, ka 480.

$$2x^{2}+28x=480$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+28x}{2}=\frac{480}{2}$$

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

$$x^{2}+\frac{28}{2}x=\frac{480}{2}$$

Whakawehe 28 ki te 2.

$$x^{2}+14x=\frac{480}{2}$$

Whakawehe 480 ki te 2.

$$x^{2}+14x=240$$

Whakawehea te 14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 7. Nā, tāpiria te pūrua o te 7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+14x+7^{2}=240+7^{2}$$

Pūrua 7.

$$x^{2}+14x+49=240+49$$

Tāpiri 240 ki te 49.

$$x^{2}+14x+49=289$$

Tauwehea x^{2}+14x+49. 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}.

$$\left(x+7\right)^{2}=289$$

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

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{289}$$

Whakarūnātia.

$$x+7=17 x+7=-17$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x=10 x=-24$$