Whakaoti i te x^2+(x+5)^2=25^2

$$x ^ { 2 } + ( x + 5 ) ^ { 2 } = 25 ^ { 2 }$$

Whakaoti mō x

$$x=-20$$

$$x=15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+5\right)^{2}.

$$x^{2}+x^{2}+10x+25=25^{2}$$

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

$$2x^{2}+10x+25=25^{2}$$

Tātaihia te 25 mā te pū o 2, kia riro ko 625.

$$2x^{2}+10x+25=625$$

Tangohia te 625 mai i ngā taha e rua.

$$2x^{2}+10x+25-625=0$$

Tangohia te 625 i te 25, ka -600.

$$2x^{2}+10x-600=0$$

Whakawehea ngā taha e rua ki te 2.

$$x^{2}+5x-300=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-300. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=1\left(-300\right)=-300$$

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 -300.

$$-1,300 -2,150 -3,100 -4,75 -5,60 -6,50 -10,30 -12,25 -15,20$$

Tātaihia te tapeke mō ia takirua.

$$-1+300=299 -2+150=148 -3+100=97 -4+75=71 -5+60=55 -6+50=44 -10+30=20 -12+25=13 -15+20=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-15 b=20$$

Tuhia anō te x^{2}+5x-300 hei \left(x^{2}-15x\right)+\left(20x-300\right).

$$\left(x^{2}-15x\right)+\left(20x-300\right)$$

Tauwehea te x i te tuatahi me te 20 i te rōpū tuarua.

$$x\left(x-15\right)+20\left(x-15\right)$$

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

$$\left(x-15\right)\left(x+20\right)$$

Hei kimi otinga whārite, me whakaoti te x-15=0 me te x+20=0.

$$x=15 x=-20$$

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(x+5\right)^{2}.

$$x^{2}+x^{2}+10x+25=25^{2}$$

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

$$2x^{2}+10x+25=25^{2}$$

Tātaihia te 25 mā te pū o 2, kia riro ko 625.

$$2x^{2}+10x+25=625$$

Tangohia te 625 mai i ngā taha e rua.

$$2x^{2}+10x+25-625=0$$

Tangohia te 625 i te 25, ka -600.

$$2x^{2}+10x-600=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 10 mō b, me -600 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 2\left(-600\right)}}{2\times 2}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 2\left(-600\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-10±\sqrt{100-8\left(-600\right)}}{2\times 2}$$

Whakareatia -8 ki te -600.

$$x=\frac{-10±\sqrt{100+4800}}{2\times 2}$$

Tāpiri 100 ki te 4800.

$$x=\frac{-10±\sqrt{4900}}{2\times 2}$$

Tuhia te pūtakerua o te 4900.

$$x=\frac{-10±70}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-10±70}{4}$$

Nā, me whakaoti te whārite x=\frac{-10±70}{4} ina he tāpiri te ±. Tāpiri -10 ki te 70.

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

Whakawehe 60 ki te 4.

$$x=15$$

Nā, me whakaoti te whārite x=\frac{-10±70}{4} ina he tango te ±. Tango 70 mai i -10.

$$x=-\frac{80}{4}$$

Whakawehe -80 ki te 4.

$$x=-20$$

Kua oti te whārite te whakatau.

$$x=15 x=-20$$

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(x+5\right)^{2}.

$$x^{2}+x^{2}+10x+25=25^{2}$$

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

$$2x^{2}+10x+25=25^{2}$$

Tātaihia te 25 mā te pū o 2, kia riro ko 625.

$$2x^{2}+10x+25=625$$

Tangohia te 25 mai i ngā taha e rua.

$$2x^{2}+10x=625-25$$

Tangohia te 25 i te 625, ka 600.

$$2x^{2}+10x=600$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+10x}{2}=\frac{600}{2}$$

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

$$x^{2}+\frac{10}{2}x=\frac{600}{2}$$

Whakawehe 10 ki te 2.

$$x^{2}+5x=\frac{600}{2}$$

Whakawehe 600 ki te 2.

$$x^{2}+5x=300$$

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.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=300+\left(\frac{5}{2}\right)^{2}$$

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

$$x^{2}+5x+\frac{25}{4}=300+\frac{25}{4}$$

Tāpiri 300 ki te \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{1225}{4}$$

Tauwehea x^{2}+5x+\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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{1225}{4}$$

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

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{1225}{4}}$$

Whakarūnātia.

$$x+\frac{5}{2}=\frac{35}{2} x+\frac{5}{2}=-\frac{35}{2}$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$x=15 x=-20$$