Whakaoti i te 4^2=x^2+(7-2x/2)^2

$$4 ^ { 2 } = x ^ { 2 } + ( \frac { 7 - 2 x } { 2 } ) ^ { 2 }$$

Whakaoti mō x

$$x = \frac{\sqrt{79} + 7}{4} \approx 3.972048604$$

$$x=\frac{7-\sqrt{79}}{4}\approx -0.472048604$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$16=x^{2}+\left(\frac{7-2x}{2}\right)^{2}$$

Kia whakarewa i te \frac{7-2x}{2} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$16=x^{2}+\frac{\left(7-2x\right)^{2}}{2^{2}}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x^{2} ki te \frac{2^{2}}{2^{2}}.

$$16=\frac{x^{2}\times 2^{2}}{2^{2}}+\frac{\left(7-2x\right)^{2}}{2^{2}}$$

Tā te mea he rite te tauraro o \frac{x^{2}\times 2^{2}}{2^{2}} me \frac{\left(7-2x\right)^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$16=\frac{x^{2}\times 2^{2}+\left(7-2x\right)^{2}}{2^{2}}$$

Mahia ngā whakarea i roto o x^{2}\times 2^{2}+\left(7-2x\right)^{2}.

$$16=\frac{4x^{2}+49-28x+4x^{2}}{2^{2}}$$

Whakakotahitia ngā kupu rite i 4x^{2}+49-28x+4x^{2}.

$$16=\frac{8x^{2}+49-28x}{2^{2}}$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$16=\frac{8x^{2}+49-28x}{4}$$

Whakawehea ia wā o 8x^{2}+49-28x ki te 4, kia riro ko 2x^{2}+\frac{49}{4}-7x.

$$16=2x^{2}+\frac{49}{4}-7x$$

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

$$2x^{2}+\frac{49}{4}-7x=16$$

Tangohia te 16 mai i ngā taha e rua.

$$2x^{2}+\frac{49}{4}-7x-16=0$$

Tangohia te 16 i te \frac{49}{4}, ka -\frac{15}{4}.

$$2x^{2}-\frac{15}{4}-7x=0$$

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.

$$2x^{2}-7x-\frac{15}{4}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -7 mō b, me -\frac{15}{4} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 2\left(-\frac{15}{4}\right)}}{2\times 2}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 2\left(-\frac{15}{4}\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-8\left(-\frac{15}{4}\right)}}{2\times 2}$$

Whakareatia -8 ki te -\frac{15}{4}.

$$x=\frac{-\left(-7\right)±\sqrt{49+30}}{2\times 2}$$

Tāpiri 49 ki te 30.

$$x=\frac{-\left(-7\right)±\sqrt{79}}{2\times 2}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±\sqrt{79}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{7±\sqrt{79}}{4}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{79}}{4} ina he tāpiri te ±. Tāpiri 7 ki te \sqrt{79}.

$$x=\frac{\sqrt{79}+7}{4}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{79}}{4} ina he tango te ±. Tango \sqrt{79} mai i 7.

$$x=\frac{7-\sqrt{79}}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{79}+7}{4} x=\frac{7-\sqrt{79}}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$16=x^{2}+\left(\frac{7-2x}{2}\right)^{2}$$

Kia whakarewa i te \frac{7-2x}{2} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$16=x^{2}+\frac{\left(7-2x\right)^{2}}{2^{2}}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x^{2} ki te \frac{2^{2}}{2^{2}}.

$$16=\frac{x^{2}\times 2^{2}}{2^{2}}+\frac{\left(7-2x\right)^{2}}{2^{2}}$$

Tā te mea he rite te tauraro o \frac{x^{2}\times 2^{2}}{2^{2}} me \frac{\left(7-2x\right)^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$16=\frac{x^{2}\times 2^{2}+\left(7-2x\right)^{2}}{2^{2}}$$

Mahia ngā whakarea i roto o x^{2}\times 2^{2}+\left(7-2x\right)^{2}.

$$16=\frac{4x^{2}+49-28x+4x^{2}}{2^{2}}$$

Whakakotahitia ngā kupu rite i 4x^{2}+49-28x+4x^{2}.

$$16=\frac{8x^{2}+49-28x}{2^{2}}$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$16=\frac{8x^{2}+49-28x}{4}$$

Whakawehea ia wā o 8x^{2}+49-28x ki te 4, kia riro ko 2x^{2}+\frac{49}{4}-7x.

$$16=2x^{2}+\frac{49}{4}-7x$$

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

$$2x^{2}+\frac{49}{4}-7x=16$$

Tangohia te \frac{49}{4} mai i ngā taha e rua.

$$2x^{2}-7x=16-\frac{49}{4}$$

Tangohia te \frac{49}{4} i te 16, ka \frac{15}{4}.

$$2x^{2}-7x=\frac{15}{4}$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-7x}{2}=\frac{\frac{15}{4}}{2}$$

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

$$x^{2}-\frac{7}{2}x=\frac{\frac{15}{4}}{2}$$

Whakawehe \frac{15}{4} ki te 2.

$$x^{2}-\frac{7}{2}x=\frac{15}{8}$$

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=\frac{15}{8}+\left(-\frac{7}{4}\right)^{2}$$

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{15}{8}+\frac{49}{16}$$

Tāpiri \frac{15}{8} ki te \frac{49}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{79}{16}$$

Tauwehea x^{2}-\frac{7}{2}x+\frac{49}{16}. 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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{79}{16}$$

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

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{79}{16}}$$

Whakarūnātia.

$$x-\frac{7}{4}=\frac{\sqrt{79}}{4} x-\frac{7}{4}=-\frac{\sqrt{79}}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{79}+7}{4} x=\frac{7-\sqrt{79}}{4}$$