Whakaoti i te (x+3)^2-x^2=x^2

$$( x + 3 ) ^ { 2 } - x ^ { 2 } = x ^ { 2 }$$

Whakaoti mō x

$$x=3\sqrt{2}+3\approx 7.242640687$$

$$x=3-3\sqrt{2}\approx -1.242640687$$

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+3\right)^{2}.

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

Pahekotia te x^{2} me -x^{2}, ka 0.

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

Tangohia te x^{2} mai i ngā taha e rua.

$$6x+9-x^{2}=0$$

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.

$$-x^{2}+6x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 6 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-6±\sqrt{36+4\times 9}}{2\left(-1\right)}$$

Whakareatia 4 ki te 9.

$$x=\frac{-6±\sqrt{36+36}}{2\left(-1\right)}$$

Tāpiri 36 ki te 36.

$$x=\frac{-6±\sqrt{72}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 72.

$$x=\frac{-6±6\sqrt{2}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-6±6\sqrt{2}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-6±6\sqrt{2}}{-2} ina he tāpiri te ±. Tāpiri -6 ki te 6\sqrt{2}.

$$x=\frac{6\sqrt{2}-6}{-2}$$

Whakawehe -6+6\sqrt{2} ki te -2.

$$x=3-3\sqrt{2}$$

Nā, me whakaoti te whārite x=\frac{-6±6\sqrt{2}}{-2} ina he tango te ±. Tango 6\sqrt{2} mai i -6.

$$x=\frac{-6\sqrt{2}-6}{-2}$$

Whakawehe -6-6\sqrt{2} ki te -2.

$$x=3\sqrt{2}+3$$

Kua oti te whārite te whakatau.

$$x=3-3\sqrt{2} x=3\sqrt{2}+3$$

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+3\right)^{2}.

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

Pahekotia te x^{2} me -x^{2}, ka 0.

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

Tangohia te x^{2} mai i ngā taha e rua.

$$6x+9-x^{2}=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

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.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+6x}{-1}=-\frac{9}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{6}{-1}x=-\frac{9}{-1}$$

Whakawehe 6 ki te -1.

$$x^{2}-6x=-\frac{9}{-1}$$

Whakawehe -9 ki te -1.

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

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.

$$x^{2}-6x+\left(-3\right)^{2}=9+\left(-3\right)^{2}$$

Pūrua -3.

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

Tāpiri 9 ki te 9.

$$x^{2}-6x+9=18$$

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}.

$$\left(x-3\right)^{2}=18$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{18}$$

Whakarūnātia.

$$x-3=3\sqrt{2} x-3=-3\sqrt{2}$$

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

$$x=3\sqrt{2}+3 x=3-3\sqrt{2}$$