Whakaoti i te r^2-23r+24=0

$${ r }^{ 2 } -23r+24=0$$

Whakaoti mō r

$$r = \frac{\sqrt{433} + 23}{2} \approx 21.904326023$$

$$r = \frac{23 - \sqrt{433}}{2} \approx 1.095673977$$

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.

$$r^{2}-23r+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -23 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 24}}{2}$$

Pūrua -23.

$$r=\frac{-\left(-23\right)±\sqrt{529-4\times 24}}{2}$$

Whakareatia -4 ki te 24.

$$r=\frac{-\left(-23\right)±\sqrt{529-96}}{2}$$

Tāpiri 529 ki te -96.

$$r=\frac{-\left(-23\right)±\sqrt{433}}{2}$$

Ko te tauaro o -23 ko 23.

$$r=\frac{23±\sqrt{433}}{2}$$

Nā, me whakaoti te whārite r=\frac{23±\sqrt{433}}{2} ina he tāpiri te ±. Tāpiri 23 ki te \sqrt{433}.

$$r=\frac{\sqrt{433}+23}{2}$$

Nā, me whakaoti te whārite r=\frac{23±\sqrt{433}}{2} ina he tango te ±. Tango \sqrt{433} mai i 23.

$$r=\frac{23-\sqrt{433}}{2}$$

Kua oti te whārite te whakatau.

$$r=\frac{\sqrt{433}+23}{2} r=\frac{23-\sqrt{433}}{2}$$

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.

$$r^{2}-23r+24=0$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$r^{2}-23r+24-24=-24$$

Mā te tango i te 24 i a ia ake anō ka toe ko te 0.

$$r^{2}-23r=-24$$

Whakawehea te -23, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{23}{2}. Nā, tāpiria te pūrua o te -\frac{23}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}-23r+\left(-\frac{23}{2}\right)^{2}=-24+\left(-\frac{23}{2}\right)^{2}$$

Pūruatia -\frac{23}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$r^{2}-23r+\frac{529}{4}=-24+\frac{529}{4}$$

Tāpiri -24 ki te \frac{529}{4}.

$$r^{2}-23r+\frac{529}{4}=\frac{433}{4}$$

Tauwehea r^{2}-23r+\frac{529}{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}.

$$\left(r-\frac{23}{2}\right)^{2}=\frac{433}{4}$$

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

$$\sqrt{\left(r-\frac{23}{2}\right)^{2}}=\sqrt{\frac{433}{4}}$$

Whakarūnātia.

$$r-\frac{23}{2}=\frac{\sqrt{433}}{2} r-\frac{23}{2}=-\frac{\sqrt{433}}{2}$$

Me tāpiri \frac{23}{2} ki ngā taha e rua o te whārite.

$$r=\frac{\sqrt{433}+23}{2} r=\frac{23-\sqrt{433}}{2}$$