Whakaoti i te 15-x^2+4x-5=0

$$15 - x ^ { 2 } + 4 x - 5 = 0$$

Whakaoti mō x

$$x=\sqrt{14}+2\approx 5.741657387$$

$$x=2-\sqrt{14}\approx -1.741657387$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 5 i te 15, ka 10.

$$10-x^{2}+4x=0$$

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.

$$-x^{2}+4x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 4 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-4±\sqrt{16+4\times 10}}{2\left(-1\right)}$$

Whakareatia 4 ki te 10.

$$x=\frac{-4±\sqrt{16+40}}{2\left(-1\right)}$$

Tāpiri 16 ki te 40.

$$x=\frac{-4±\sqrt{56}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 56.

$$x=\frac{-4±2\sqrt{14}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-4±2\sqrt{14}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{14}}{-2} ina he tāpiri te ±. Tāpiri -4 ki te 2\sqrt{14}.

$$x=\frac{2\sqrt{14}-4}{-2}$$

Whakawehe -4+2\sqrt{14} ki te -2.

$$x=2-\sqrt{14}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{14}}{-2} ina he tango te ±. Tango 2\sqrt{14} mai i -4.

$$x=\frac{-2\sqrt{14}-4}{-2}$$

Whakawehe -4-2\sqrt{14} ki te -2.

$$x=\sqrt{14}+2$$

Kua oti te whārite te whakatau.

$$x=2-\sqrt{14} x=\sqrt{14}+2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 5 i te 15, ka 10.

$$10-x^{2}+4x=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-x^{2}+4x=-10$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+4x}{-1}=-\frac{10}{-1}$$

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

$$x^{2}+\frac{4}{-1}x=-\frac{10}{-1}$$

Whakawehe 4 ki te -1.

$$x^{2}-4x=-\frac{10}{-1}$$

Whakawehe -10 ki te -1.

$$x^{2}-4x=10$$

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

$$x^{2}-4x+\left(-2\right)^{2}=10+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=10+4$$

Tāpiri 10 ki te 4.

$$x^{2}-4x+4=14$$

Tauwehea x^{2}-4x+4. 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}.

$$\left(x-2\right)^{2}=14$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{14}$$

Whakarūnātia.

$$x-2=\sqrt{14} x-2=-\sqrt{14}$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=\sqrt{14}+2 x=2-\sqrt{14}$$