Whakaoti i te 7r^2+40r-700=0

$$7 r ^ { 2 } + 40 r - 700 = 0$$

Whakaoti mō r

$$r = \frac{10 \sqrt{53} - 20}{7} \approx 7.543014128$$

$$r=\frac{-10\sqrt{53}-20}{7}\approx -13.257299842$$

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.

$$7r^{2}+40r-700=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 40 mō b, me -700 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-40±\sqrt{40^{2}-4\times 7\left(-700\right)}}{2\times 7}$$

Pūrua 40.

$$r=\frac{-40±\sqrt{1600-4\times 7\left(-700\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$r=\frac{-40±\sqrt{1600-28\left(-700\right)}}{2\times 7}$$

Whakareatia -28 ki te -700.

$$r=\frac{-40±\sqrt{1600+19600}}{2\times 7}$$

Tāpiri 1600 ki te 19600.

$$r=\frac{-40±\sqrt{21200}}{2\times 7}$$

Tuhia te pūtakerua o te 21200.

$$r=\frac{-40±20\sqrt{53}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$r=\frac{-40±20\sqrt{53}}{14}$$

Nā, me whakaoti te whārite r=\frac{-40±20\sqrt{53}}{14} ina he tāpiri te ±. Tāpiri -40 ki te 20\sqrt{53}.

$$r=\frac{20\sqrt{53}-40}{14}$$

Whakawehe -40+20\sqrt{53} ki te 14.

$$r=\frac{10\sqrt{53}-20}{7}$$

Nā, me whakaoti te whārite r=\frac{-40±20\sqrt{53}}{14} ina he tango te ±. Tango 20\sqrt{53} mai i -40.

$$r=\frac{-20\sqrt{53}-40}{14}$$

Whakawehe -40-20\sqrt{53} ki te 14.

$$r=\frac{-10\sqrt{53}-20}{7}$$

Kua oti te whārite te whakatau.

$$r=\frac{10\sqrt{53}-20}{7} r=\frac{-10\sqrt{53}-20}{7}$$

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.

$$7r^{2}+40r-700=0$$

Me tāpiri 700 ki ngā taha e rua o te whārite.

$$7r^{2}+40r-700-\left(-700\right)=-\left(-700\right)$$

Mā te tango i te -700 i a ia ake anō ka toe ko te 0.

$$7r^{2}+40r=-\left(-700\right)$$

Tango -700 mai i 0.

$$7r^{2}+40r=700$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7r^{2}+40r}{7}=\frac{700}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$r^{2}+\frac{40}{7}r=\frac{700}{7}$$

Whakawehe 700 ki te 7.

$$r^{2}+\frac{40}{7}r=100$$

Whakawehea te \frac{40}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{20}{7}. Nā, tāpiria te pūrua o te \frac{20}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}+\frac{40}{7}r+\left(\frac{20}{7}\right)^{2}=100+\left(\frac{20}{7}\right)^{2}$$

Pūruatia \frac{20}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$r^{2}+\frac{40}{7}r+\frac{400}{49}=100+\frac{400}{49}$$

Tāpiri 100 ki te \frac{400}{49}.

$$r^{2}+\frac{40}{7}r+\frac{400}{49}=\frac{5300}{49}$$

Tauwehea r^{2}+\frac{40}{7}r+\frac{400}{49}. 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}.

$$\left(r+\frac{20}{7}\right)^{2}=\frac{5300}{49}$$

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

$$\sqrt{\left(r+\frac{20}{7}\right)^{2}}=\sqrt{\frac{5300}{49}}$$

Whakarūnātia.

$$r+\frac{20}{7}=\frac{10\sqrt{53}}{7} r+\frac{20}{7}=-\frac{10\sqrt{53}}{7}$$

Me tango \frac{20}{7} mai i ngā taha e rua o te whārite.

$$r=\frac{10\sqrt{53}-20}{7} r=\frac{-10\sqrt{53}-20}{7}$$