Whakaoti i te x^2+(x+3/2)^2-8x-2(x+3/2)+7=0

$$x ^ { 2 } + ( \frac { x + 3 } { 2 } ) ^ { 2 } - 8 x - 2 ( \frac { x + 3 } { 2 } ) + 7 = 0$$

Whakaoti mō x

$$x=1$$

$$x=5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2\left(x^{2}+\left(\frac{x+3}{2}\right)^{2}-8x-2\times \frac{x+3}{2}\right)+14=0$$

Kia whakarewa i te \frac{x+3}{2} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$2\left(x^{2}+\frac{\left(x+3\right)^{2}}{2^{2}}-8x-2\times \frac{x+3}{2}\right)+14=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x^{2}-8x ki te \frac{2^{2}}{2^{2}}.

$$2\left(\frac{\left(x^{2}-8x\right)\times 2^{2}}{2^{2}}+\frac{\left(x+3\right)^{2}}{2^{2}}-2\times \frac{x+3}{2}\right)+14=0$$

Tā te mea he rite te tauraro o \frac{\left(x^{2}-8x\right)\times 2^{2}}{2^{2}} me \frac{\left(x+3\right)^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$2\left(\frac{\left(x^{2}-8x\right)\times 2^{2}+\left(x+3\right)^{2}}{2^{2}}-2\times \frac{x+3}{2}\right)+14=0$$

Mahia ngā whakarea i roto o \left(x^{2}-8x\right)\times 2^{2}+\left(x+3\right)^{2}.

$$2\left(\frac{4x^{2}-32x+x^{2}+6x+9}{2^{2}}-2\times \frac{x+3}{2}\right)+14=0$$

Whakakotahitia ngā kupu rite i 4x^{2}-32x+x^{2}+6x+9.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}-2\times \frac{x+3}{2}\right)+14=0$$

Tuhia te 2\times \frac{x+3}{2} hei hautanga kotahi.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}-\frac{2\left(x+3\right)}{2}\right)+14=0$$

Me whakakore te 2 me te 2.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}-\left(x+3\right)\right)+14=0$$

Hei kimi i te tauaro o x+3, kimihia te tauaro o ia taurangi.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}-x-3\right)+14=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia -x-3 ki te \frac{2^{2}}{2^{2}}.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}+\frac{\left(-x-3\right)\times 2^{2}}{2^{2}}\right)+14=0$$

Tā te mea he rite te tauraro o \frac{5x^{2}-26x+9}{2^{2}} me \frac{\left(-x-3\right)\times 2^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$2\times \frac{5x^{2}-26x+9+\left(-x-3\right)\times 2^{2}}{2^{2}}+14=0$$

Mahia ngā whakarea i roto o 5x^{2}-26x+9+\left(-x-3\right)\times 2^{2}.

$$2\times \frac{5x^{2}-26x+9-4x-12}{2^{2}}+14=0$$

Whakakotahitia ngā kupu rite i 5x^{2}-26x+9-4x-12.

$$2\times \frac{5x^{2}-30x-3}{2^{2}}+14=0$$

Tuhia te 2\times \frac{5x^{2}-30x-3}{2^{2}} hei hautanga kotahi.

$$\frac{2\left(5x^{2}-30x-3\right)}{2^{2}}+14=0$$

Me whakakore tahi te 2 i te taurunga me te tauraro.

$$\frac{5x^{2}-30x-3}{2}+14=0$$

Whakawehea ia wā o 5x^{2}-30x-3 ki te 2, kia riro ko \frac{5}{2}x^{2}-15x-\frac{3}{2}.

$$\frac{5}{2}x^{2}-15x-\frac{3}{2}+14=0$$

Tāpirihia te -\frac{3}{2} ki te 14, ka \frac{25}{2}.

$$\frac{5}{2}x^{2}-15x+\frac{25}{2}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{5}{2} mō a, -15 mō b, me \frac{25}{2} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times \frac{5}{2}\times \frac{25}{2}}}{2\times \frac{5}{2}}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times \frac{5}{2}\times \frac{25}{2}}}{2\times \frac{5}{2}}$$

Whakareatia -4 ki te \frac{5}{2}.

$$x=\frac{-\left(-15\right)±\sqrt{225-10\times \frac{25}{2}}}{2\times \frac{5}{2}}$$

Whakareatia -10 ki te \frac{25}{2}.

$$x=\frac{-\left(-15\right)±\sqrt{225-125}}{2\times \frac{5}{2}}$$

Tāpiri 225 ki te -125.

$$x=\frac{-\left(-15\right)±\sqrt{100}}{2\times \frac{5}{2}}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-15\right)±10}{2\times \frac{5}{2}}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±10}{2\times \frac{5}{2}}$$

Whakareatia 2 ki te \frac{5}{2}.

$$x=\frac{15±10}{5}$$

Nā, me whakaoti te whārite x=\frac{15±10}{5} ina he tāpiri te ±. Tāpiri 15 ki te 10.

$$x=\frac{25}{5}$$

Whakawehe 25 ki te 5.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{15±10}{5} ina he tango te ±. Tango 10 mai i 15.

$$x=\frac{5}{5}$$

Whakawehe 5 ki te 5.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=5 x=1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2\left(x^{2}+\left(\frac{x+3}{2}\right)^{2}-8x-2\times \frac{x+3}{2}\right)+14=0$$

Kia whakarewa i te \frac{x+3}{2} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$2\left(x^{2}+\frac{\left(x+3\right)^{2}}{2^{2}}-8x-2\times \frac{x+3}{2}\right)+14=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x^{2}-8x ki te \frac{2^{2}}{2^{2}}.

$$2\left(\frac{\left(x^{2}-8x\right)\times 2^{2}}{2^{2}}+\frac{\left(x+3\right)^{2}}{2^{2}}-2\times \frac{x+3}{2}\right)+14=0$$

Tā te mea he rite te tauraro o \frac{\left(x^{2}-8x\right)\times 2^{2}}{2^{2}} me \frac{\left(x+3\right)^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$2\left(\frac{\left(x^{2}-8x\right)\times 2^{2}+\left(x+3\right)^{2}}{2^{2}}-2\times \frac{x+3}{2}\right)+14=0$$

Mahia ngā whakarea i roto o \left(x^{2}-8x\right)\times 2^{2}+\left(x+3\right)^{2}.

$$2\left(\frac{4x^{2}-32x+x^{2}+6x+9}{2^{2}}-2\times \frac{x+3}{2}\right)+14=0$$

Whakakotahitia ngā kupu rite i 4x^{2}-32x+x^{2}+6x+9.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}-2\times \frac{x+3}{2}\right)+14=0$$

Tuhia te 2\times \frac{x+3}{2} hei hautanga kotahi.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}-\frac{2\left(x+3\right)}{2}\right)+14=0$$

Me whakakore te 2 me te 2.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}-\left(x+3\right)\right)+14=0$$

Hei kimi i te tauaro o x+3, kimihia te tauaro o ia taurangi.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}-x-3\right)+14=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia -x-3 ki te \frac{2^{2}}{2^{2}}.

$$2\left(\frac{5x^{2}-26x+9}{2^{2}}+\frac{\left(-x-3\right)\times 2^{2}}{2^{2}}\right)+14=0$$

Tā te mea he rite te tauraro o \frac{5x^{2}-26x+9}{2^{2}} me \frac{\left(-x-3\right)\times 2^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$2\times \frac{5x^{2}-26x+9+\left(-x-3\right)\times 2^{2}}{2^{2}}+14=0$$

Mahia ngā whakarea i roto o 5x^{2}-26x+9+\left(-x-3\right)\times 2^{2}.

$$2\times \frac{5x^{2}-26x+9-4x-12}{2^{2}}+14=0$$

Whakakotahitia ngā kupu rite i 5x^{2}-26x+9-4x-12.

$$2\times \frac{5x^{2}-30x-3}{2^{2}}+14=0$$

Tuhia te 2\times \frac{5x^{2}-30x-3}{2^{2}} hei hautanga kotahi.

$$\frac{2\left(5x^{2}-30x-3\right)}{2^{2}}+14=0$$

Me whakakore tahi te 2 i te taurunga me te tauraro.

$$\frac{5x^{2}-30x-3}{2}+14=0$$

Whakawehea ia wā o 5x^{2}-30x-3 ki te 2, kia riro ko \frac{5}{2}x^{2}-15x-\frac{3}{2}.

$$\frac{5}{2}x^{2}-15x-\frac{3}{2}+14=0$$

Tāpirihia te -\frac{3}{2} ki te 14, ka \frac{25}{2}.

$$\frac{5}{2}x^{2}-15x+\frac{25}{2}=0$$

Tangohia te \frac{25}{2} mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$\frac{5}{2}x^{2}-15x=-\frac{25}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{5}{2}x^{2}-15x}{\frac{5}{2}}=-\frac{\frac{25}{2}}{\frac{5}{2}}$$

Mā te whakawehe ki te \frac{5}{2} ka wetekia te whakareanga ki te \frac{5}{2}.

$$x^{2}+\left(-\frac{15}{\frac{5}{2}}\right)x=-\frac{\frac{25}{2}}{\frac{5}{2}}$$

Whakawehe -15 ki te \frac{5}{2} mā te whakarea -15 ki te tau huripoki o \frac{5}{2}.

$$x^{2}-6x=-\frac{\frac{25}{2}}{\frac{5}{2}}$$

Whakawehe -\frac{25}{2} ki te \frac{5}{2} mā te whakarea -\frac{25}{2} ki te tau huripoki o \frac{5}{2}.

$$x^{2}-6x=-5$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=-5+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-5+9$$

Tāpiri -5 ki te 9.

$$x^{2}-6x+9=4$$

Tauwehea x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=4$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x-3=2 x-3=-2$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=5 x=1$$