Whakaoti i te -1/2left(2-xright)^2+4+2x+6=x

$$- \frac{ 1 }{ 2 } { \left(2-x \right) }^{ 2 } +4+2x+6=x$$

Whakaoti mō x

$$x=-2$$

$$x=8$$

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(2-x\right)^{2}.

$$-\frac{1}{2}\left(4-4x+x^{2}\right)+4+2x+6=x$$

Whakamahia te āhuatanga tohatoha hei whakarea te -\frac{1}{2} ki te 4-4x+x^{2}.

$$-2+2x-\frac{1}{2}x^{2}+4+2x+6=x$$

Tāpirihia te -2 ki te 4, ka 2.

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

Pahekotia te 2x me 2x, ka 4x.

$$2+4x-\frac{1}{2}x^{2}+6=x$$

Tāpirihia te 2 ki te 6, ka 8.

$$8+4x-\frac{1}{2}x^{2}=x$$

Tangohia te x mai i ngā taha e rua.

$$8+4x-\frac{1}{2}x^{2}-x=0$$

Pahekotia te 4x me -x, ka 3x.

$$8+3x-\frac{1}{2}x^{2}=0$$

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.

$$-\frac{1}{2}x^{2}+3x+8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{2} mō a, 3 mō b, me 8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-\frac{1}{2}\right)\times 8}}{2\left(-\frac{1}{2}\right)}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\left(-\frac{1}{2}\right)\times 8}}{2\left(-\frac{1}{2}\right)}$$

Whakareatia -4 ki te -\frac{1}{2}.

$$x=\frac{-3±\sqrt{9+2\times 8}}{2\left(-\frac{1}{2}\right)}$$

Whakareatia 2 ki te 8.

$$x=\frac{-3±\sqrt{9+16}}{2\left(-\frac{1}{2}\right)}$$

Tāpiri 9 ki te 16.

$$x=\frac{-3±\sqrt{25}}{2\left(-\frac{1}{2}\right)}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-3±5}{2\left(-\frac{1}{2}\right)}$$

Whakareatia 2 ki te -\frac{1}{2}.

$$x=\frac{-3±5}{-1}$$

Nā, me whakaoti te whārite x=\frac{-3±5}{-1} ina he tāpiri te ±. Tāpiri -3 ki te 5.

$$x=\frac{2}{-1}$$

Whakawehe 2 ki te -1.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-3±5}{-1} ina he tango te ±. Tango 5 mai i -3.

$$x=-\frac{8}{-1}$$

Whakawehe -8 ki te -1.

$$x=8$$

Kua oti te whārite te whakatau.

$$x=-2 x=8$$

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(2-x\right)^{2}.

$$-\frac{1}{2}\left(4-4x+x^{2}\right)+4+2x+6=x$$

Whakamahia te āhuatanga tohatoha hei whakarea te -\frac{1}{2} ki te 4-4x+x^{2}.

$$-2+2x-\frac{1}{2}x^{2}+4+2x+6=x$$

Tāpirihia te -2 ki te 4, ka 2.

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

Pahekotia te 2x me 2x, ka 4x.

$$2+4x-\frac{1}{2}x^{2}+6=x$$

Tāpirihia te 2 ki te 6, ka 8.

$$8+4x-\frac{1}{2}x^{2}=x$$

Tangohia te x mai i ngā taha e rua.

$$8+4x-\frac{1}{2}x^{2}-x=0$$

Pahekotia te 4x me -x, ka 3x.

$$8+3x-\frac{1}{2}x^{2}=0$$

Tangohia te 8 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-\frac{1}{2}x^{2}=-8$$

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.

$$-\frac{1}{2}x^{2}+3x=-8$$

Me whakarea ngā taha e rua ki te -2.

$$\frac{-\frac{1}{2}x^{2}+3x}{-\frac{1}{2}}=-\frac{8}{-\frac{1}{2}}$$

Mā te whakawehe ki te -\frac{1}{2} ka wetekia te whakareanga ki te -\frac{1}{2}.

$$x^{2}+\frac{3}{-\frac{1}{2}}x=-\frac{8}{-\frac{1}{2}}$$

Whakawehe 3 ki te -\frac{1}{2} mā te whakarea 3 ki te tau huripoki o -\frac{1}{2}.

$$x^{2}-6x=-\frac{8}{-\frac{1}{2}}$$

Whakawehe -8 ki te -\frac{1}{2} mā te whakarea -8 ki te tau huripoki o -\frac{1}{2}.

$$x^{2}-6x=16$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=16+9$$

Tāpiri 16 ki te 9.

$$x^{2}-6x+9=25$$

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}.

$$\left(x-3\right)^{2}=25$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x-3=5 x-3=-5$$

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

$$x=8 x=-2$$