Whakaoti i te -9x^2+72x-102=0

$$-9 { x }^{ 2 } +72x-102 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{42}}{3}+4\approx 6.160246899$$

$$x=-\frac{\sqrt{42}}{3}+4\approx 1.839753101$$

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.

$$-9x^{2}+72x-102=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -9 mō a, 72 mō b, me -102 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\left(-9\right)\left(-102\right)}}{2\left(-9\right)}$$

Pūrua 72.

$$x=\frac{-72±\sqrt{5184-4\left(-9\right)\left(-102\right)}}{2\left(-9\right)}$$

Whakareatia -4 ki te -9.

$$x=\frac{-72±\sqrt{5184+36\left(-102\right)}}{2\left(-9\right)}$$

Whakareatia 36 ki te -102.

$$x=\frac{-72±\sqrt{5184-3672}}{2\left(-9\right)}$$

Tāpiri 5184 ki te -3672.

$$x=\frac{-72±\sqrt{1512}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te 1512.

$$x=\frac{-72±6\sqrt{42}}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$x=\frac{-72±6\sqrt{42}}{-18}$$

Nā, me whakaoti te whārite x=\frac{-72±6\sqrt{42}}{-18} ina he tāpiri te ±. Tāpiri -72 ki te 6\sqrt{42}.

$$x=\frac{6\sqrt{42}-72}{-18}$$

Whakawehe -72+6\sqrt{42} ki te -18.

$$x=-\frac{\sqrt{42}}{3}+4$$

Nā, me whakaoti te whārite x=\frac{-72±6\sqrt{42}}{-18} ina he tango te ±. Tango 6\sqrt{42} mai i -72.

$$x=\frac{-6\sqrt{42}-72}{-18}$$

Whakawehe -72-6\sqrt{42} ki te -18.

$$x=\frac{\sqrt{42}}{3}+4$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{42}}{3}+4 x=\frac{\sqrt{42}}{3}+4$$

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.

$$-9x^{2}+72x-102=0$$

Me tāpiri 102 ki ngā taha e rua o te whārite.

$$-9x^{2}+72x-102-\left(-102\right)=-\left(-102\right)$$

Mā te tango i te -102 i a ia ake anō ka toe ko te 0.

$$-9x^{2}+72x=-\left(-102\right)$$

Tango -102 mai i 0.

$$-9x^{2}+72x=102$$

Whakawehea ngā taha e rua ki te -9.

$$\frac{-9x^{2}+72x}{-9}=\frac{102}{-9}$$

Mā te whakawehe ki te -9 ka wetekia te whakareanga ki te -9.

$$x^{2}+\frac{72}{-9}x=\frac{102}{-9}$$

Whakawehe 72 ki te -9.

$$x^{2}-8x=\frac{102}{-9}$$

Whakahekea te hautanga \frac{102}{-9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-8x=-\frac{34}{3}$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=-\frac{34}{3}+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-\frac{34}{3}+16$$

Tāpiri -\frac{34}{3} ki te 16.

$$x^{2}-8x+16=\frac{14}{3}$$

Tauwehea x^{2}-8x+16. 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}.

$$\left(x-4\right)^{2}=\frac{14}{3}$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{\frac{14}{3}}$$

Whakarūnātia.

$$x-4=\frac{\sqrt{42}}{3} x-4=-\frac{\sqrt{42}}{3}$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{42}}{3}+4 x=-\frac{\sqrt{42}}{3}+4$$