Whakaoti i te 5x+12=x^2

$$5 x + 12 = x ^ { 2 }$$

Whakaoti mō x

$$x = \frac{\sqrt{73} + 5}{2} \approx 6.772001873$$

$$x=\frac{5-\sqrt{73}}{2}\approx -1.772001873$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$5x+12-x^{2}=0$$

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.

$$-x^{2}+5x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 5 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-5±\sqrt{25+4\times 12}}{2\left(-1\right)}$$

Whakareatia 4 ki te 12.

$$x=\frac{-5±\sqrt{25+48}}{2\left(-1\right)}$$

Tāpiri 25 ki te 48.

$$x=\frac{-5±\sqrt{73}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-5±\sqrt{73}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{73}}{-2} ina he tāpiri te ±. Tāpiri -5 ki te \sqrt{73}.

$$x=\frac{\sqrt{73}-5}{-2}$$

Whakawehe -5+\sqrt{73} ki te -2.

$$x=\frac{5-\sqrt{73}}{2}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{73}}{-2} ina he tango te ±. Tango \sqrt{73} mai i -5.

$$x=\frac{-\sqrt{73}-5}{-2}$$

Whakawehe -5-\sqrt{73} ki te -2.

$$x=\frac{\sqrt{73}+5}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{5-\sqrt{73}}{2} x=\frac{\sqrt{73}+5}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$5x+12-x^{2}=0$$

Tangohia te 12 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x-x^{2}=-12$$

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.

$$-x^{2}+5x=-12$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+5x}{-1}=-\frac{12}{-1}$$

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

$$x^{2}+\frac{5}{-1}x=-\frac{12}{-1}$$

Whakawehe 5 ki te -1.

$$x^{2}-5x=-\frac{12}{-1}$$

Whakawehe -12 ki te -1.

$$x^{2}-5x=12$$

Whakawehea te -5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{2}. Nā, tāpiria te pūrua o te -\frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=12+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=12+\frac{25}{4}$$

Tāpiri 12 ki te \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{73}{4}$$

Tauwehea x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{73}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{73}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{\sqrt{73}}{2} x-\frac{5}{2}=-\frac{\sqrt{73}}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{73}+5}{2} x=\frac{5-\sqrt{73}}{2}$$