Whakaoti i te x^2-5x-9=0

$${ x }^{ 2 } -5x-9=0$$

Whakaoti mō x

$$x = \frac{\sqrt{61} + 5}{2} \approx 6.405124838$$

$$x=\frac{5-\sqrt{61}}{2}\approx -1.405124838$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}-5x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -5 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-9\right)}}{2}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-9\right)}}{2}$$

Whakareatia -4 ki te -9.

$$x=\frac{-\left(-5\right)±\sqrt{25+36}}{2}$$

Tāpiri 25 ki te 36.

$$x=\frac{-\left(-5\right)±\sqrt{61}}{2}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±\sqrt{61}}{2}$$

Nā, me whakaoti te whārite x=\frac{5±\sqrt{61}}{2} ina he tāpiri te ±. Tāpiri 5 ki te \sqrt{61}.

$$x=\frac{\sqrt{61}+5}{2}$$

Nā, me whakaoti te whārite x=\frac{5±\sqrt{61}}{2} ina he tango te ±. Tango \sqrt{61} mai i 5.

$$x=\frac{5-\sqrt{61}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{61}+5}{2} x=\frac{5-\sqrt{61}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x^{2}-5x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x^{2}-5x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$x^{2}-5x=-\left(-9\right)$$

Tango -9 mai i 0.

$$x^{2}-5x=9$$

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.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=9+\left(-\frac{5}{2}\right)^{2}$$

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

$$x^{2}-5x+\frac{25}{4}=9+\frac{25}{4}$$

Tāpiri 9 ki te \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{61}{4}$$

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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{61}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{61}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{\sqrt{61}}{2} x-\frac{5}{2}=-\frac{\sqrt{61}}{2}$$

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

$$x=\frac{\sqrt{61}+5}{2} x=\frac{5-\sqrt{61}}{2}$$