Whakaoti i te 1-2(x-3)(x-11)=0

$$1 - 2 ( x - 3 ) ( x - 11 ) = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{66}}{2}+7\approx 11.062019202$$

$$x=-\frac{\sqrt{66}}{2}+7\approx 2.937980798$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te -1 ki te 2, ka -2.

$$1-2\left(x-3\right)\left(x-11\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x-3.

$$1+\left(-2x+6\right)\left(x-11\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -2x+6 ki te x-11 ka whakakotahi i ngā kupu rite.

$$1-2x^{2}+28x-66=0$$

Tangohia te 66 i te 1, ka -65.

$$-65-2x^{2}+28x=0$$

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.

$$-2x^{2}+28x-65=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 28 mō b, me -65 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-2\right)\left(-65\right)}}{2\left(-2\right)}$$

Pūrua 28.

$$x=\frac{-28±\sqrt{784-4\left(-2\right)\left(-65\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-28±\sqrt{784+8\left(-65\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -65.

$$x=\frac{-28±\sqrt{784-520}}{2\left(-2\right)}$$

Tāpiri 784 ki te -520.

$$x=\frac{-28±\sqrt{264}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 264.

$$x=\frac{-28±2\sqrt{66}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-28±2\sqrt{66}}{-4}$$

Nā, me whakaoti te whārite x=\frac{-28±2\sqrt{66}}{-4} ina he tāpiri te ±. Tāpiri -28 ki te 2\sqrt{66}.

$$x=\frac{2\sqrt{66}-28}{-4}$$

Whakawehe -28+2\sqrt{66} ki te -4.

$$x=-\frac{\sqrt{66}}{2}+7$$

Nā, me whakaoti te whārite x=\frac{-28±2\sqrt{66}}{-4} ina he tango te ±. Tango 2\sqrt{66} mai i -28.

$$x=\frac{-2\sqrt{66}-28}{-4}$$

Whakawehe -28-2\sqrt{66} ki te -4.

$$x=\frac{\sqrt{66}}{2}+7$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{66}}{2}+7 x=\frac{\sqrt{66}}{2}+7$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te -1 ki te 2, ka -2.

$$1-2\left(x-3\right)\left(x-11\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x-3.

$$1+\left(-2x+6\right)\left(x-11\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -2x+6 ki te x-11 ka whakakotahi i ngā kupu rite.

$$1-2x^{2}+28x-66=0$$

Tangohia te 66 i te 1, ka -65.

$$-65-2x^{2}+28x=0$$

Me tāpiri te 65 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-2x^{2}+28x=65$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+28x}{-2}=\frac{65}{-2}$$

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

$$x^{2}+\frac{28}{-2}x=\frac{65}{-2}$$

Whakawehe 28 ki te -2.

$$x^{2}-14x=\frac{65}{-2}$$

Whakawehe 65 ki te -2.

$$x^{2}-14x=-\frac{65}{2}$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-\frac{65}{2}+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-\frac{65}{2}+49$$

Tāpiri -\frac{65}{2} ki te 49.

$$x^{2}-14x+49=\frac{33}{2}$$

Tauwehea x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=\frac{33}{2}$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{\frac{33}{2}}$$

Whakarūnātia.

$$x-7=\frac{\sqrt{66}}{2} x-7=-\frac{\sqrt{66}}{2}$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{66}}{2}+7 x=-\frac{\sqrt{66}}{2}+7$$