Whakaoti i te r^2-22r-7=0

$$r ^ { 2 } - 22 r - 7 = 0$$

Whakaoti mō r

$$r=8\sqrt{2}+11\approx 22.313708499$$

$$r=11-8\sqrt{2}\approx -0.313708499$$

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.

$$r^{2}-22r-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -22 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-7\right)}}{2}$$

Pūrua -22.

$$r=\frac{-\left(-22\right)±\sqrt{484-4\left(-7\right)}}{2}$$

Whakareatia -4 ki te -7.

$$r=\frac{-\left(-22\right)±\sqrt{484+28}}{2}$$

Tāpiri 484 ki te 28.

$$r=\frac{-\left(-22\right)±\sqrt{512}}{2}$$

Tuhia te pūtakerua o te 512.

$$r=\frac{-\left(-22\right)±16\sqrt{2}}{2}$$

Ko te tauaro o -22 ko 22.

$$r=\frac{22±16\sqrt{2}}{2}$$

Nā, me whakaoti te whārite r=\frac{22±16\sqrt{2}}{2} ina he tāpiri te ±. Tāpiri 22 ki te 16\sqrt{2}.

$$r=\frac{16\sqrt{2}+22}{2}$$

Whakawehe 22+16\sqrt{2} ki te 2.

$$r=8\sqrt{2}+11$$

Nā, me whakaoti te whārite r=\frac{22±16\sqrt{2}}{2} ina he tango te ±. Tango 16\sqrt{2} mai i 22.

$$r=\frac{22-16\sqrt{2}}{2}$$

Whakawehe 22-16\sqrt{2} ki te 2.

$$r=11-8\sqrt{2}$$

Kua oti te whārite te whakatau.

$$r=8\sqrt{2}+11 r=11-8\sqrt{2}$$

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.

$$r^{2}-22r-7=0$$

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

$$r^{2}-22r-7-\left(-7\right)=-\left(-7\right)$$

Mā te tango i te -7 i a ia ake anō ka toe ko te 0.

$$r^{2}-22r=-\left(-7\right)$$

Tango -7 mai i 0.

$$r^{2}-22r=7$$

Whakawehea te -22, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -11. Nā, tāpiria te pūrua o te -11 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}-22r+\left(-11\right)^{2}=7+\left(-11\right)^{2}$$

Pūrua -11.

$$r^{2}-22r+121=7+121$$

Tāpiri 7 ki te 121.

$$r^{2}-22r+121=128$$

Tauwehea r^{2}-22r+121. 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}.

$$\left(r-11\right)^{2}=128$$

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

$$\sqrt{\left(r-11\right)^{2}}=\sqrt{128}$$

Whakarūnātia.

$$r-11=8\sqrt{2} r-11=-8\sqrt{2}$$

Me tāpiri 11 ki ngā taha e rua o te whārite.

$$r=8\sqrt{2}+11 r=11-8\sqrt{2}$$