Whakaoti i te -3r^2+90r=93

$$- 3 r ^ { 2 } + 90 r = 93$$

Whakaoti mō r

$$r=\sqrt{194}+15\approx 28.928388277$$

$$r=15-\sqrt{194}\approx 1.071611723$$

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.

$$-3r^{2}+90r=93$$

Me tango 93 mai i ngā taha e rua o te whārite.

$$-3r^{2}+90r-93=93-93$$

Mā te tango i te 93 i a ia ake anō ka toe ko te 0.

$$-3r^{2}+90r-93=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 90 mō b, me -93 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-90±\sqrt{90^{2}-4\left(-3\right)\left(-93\right)}}{2\left(-3\right)}$$

Pūrua 90.

$$r=\frac{-90±\sqrt{8100-4\left(-3\right)\left(-93\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$r=\frac{-90±\sqrt{8100+12\left(-93\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -93.

$$r=\frac{-90±\sqrt{8100-1116}}{2\left(-3\right)}$$

Tāpiri 8100 ki te -1116.

$$r=\frac{-90±\sqrt{6984}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 6984.

$$r=\frac{-90±6\sqrt{194}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$r=\frac{-90±6\sqrt{194}}{-6}$$

Nā, me whakaoti te whārite r=\frac{-90±6\sqrt{194}}{-6} ina he tāpiri te ±. Tāpiri -90 ki te 6\sqrt{194}.

$$r=\frac{6\sqrt{194}-90}{-6}$$

Whakawehe -90+6\sqrt{194} ki te -6.

$$r=15-\sqrt{194}$$

Nā, me whakaoti te whārite r=\frac{-90±6\sqrt{194}}{-6} ina he tango te ±. Tango 6\sqrt{194} mai i -90.

$$r=\frac{-6\sqrt{194}-90}{-6}$$

Whakawehe -90-6\sqrt{194} ki te -6.

$$r=\sqrt{194}+15$$

Kua oti te whārite te whakatau.

$$r=15-\sqrt{194} r=\sqrt{194}+15$$

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.

$$-3r^{2}+90r=93$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3r^{2}+90r}{-3}=\frac{93}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$r^{2}+\frac{90}{-3}r=\frac{93}{-3}$$

Whakawehe 90 ki te -3.

$$r^{2}-30r=\frac{93}{-3}$$

Whakawehe 93 ki te -3.

$$r^{2}-30r=-31$$

Whakawehea te -30, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -15. Nā, tāpiria te pūrua o te -15 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}-30r+\left(-15\right)^{2}=-31+\left(-15\right)^{2}$$

Pūrua -15.

$$r^{2}-30r+225=-31+225$$

Tāpiri -31 ki te 225.

$$r^{2}-30r+225=194$$

Tauwehea r^{2}-30r+225. 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}.

$$\left(r-15\right)^{2}=194$$

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

$$\sqrt{\left(r-15\right)^{2}}=\sqrt{194}$$

Whakarūnātia.

$$r-15=\sqrt{194} r-15=-\sqrt{194}$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$r=\sqrt{194}+15 r=15-\sqrt{194}$$