Whakaoti i te 24a^2-60a+352=0

$$24 a ^ { 2 } - 60 a + 352 = 0$$

Whakaoti mō a

$$a=\frac{\sqrt{1887}i}{12}+\frac{5}{4}\approx 1.25+3.619967771i$$

$$a=-\frac{\sqrt{1887}i}{12}+\frac{5}{4}\approx 1.25-3.619967771i$$

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.

$$24a^{2}-60a+352=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 24 mō a, -60 mō b, me 352 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 24\times 352}}{2\times 24}$$

Pūrua -60.

$$a=\frac{-\left(-60\right)±\sqrt{3600-4\times 24\times 352}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$a=\frac{-\left(-60\right)±\sqrt{3600-96\times 352}}{2\times 24}$$

Whakareatia -96 ki te 352.

$$a=\frac{-\left(-60\right)±\sqrt{3600-33792}}{2\times 24}$$

Tāpiri 3600 ki te -33792.

$$a=\frac{-\left(-60\right)±\sqrt{-30192}}{2\times 24}$$

Tuhia te pūtakerua o te -30192.

$$a=\frac{-\left(-60\right)±4\sqrt{1887}i}{2\times 24}$$

Ko te tauaro o -60 ko 60.

$$a=\frac{60±4\sqrt{1887}i}{2\times 24}$$

Whakareatia 2 ki te 24.

$$a=\frac{60±4\sqrt{1887}i}{48}$$

Nā, me whakaoti te whārite a=\frac{60±4\sqrt{1887}i}{48} ina he tāpiri te ±. Tāpiri 60 ki te 4i\sqrt{1887}.

$$a=\frac{60+4\sqrt{1887}i}{48}$$

Whakawehe 60+4i\sqrt{1887} ki te 48.

$$a=\frac{\sqrt{1887}i}{12}+\frac{5}{4}$$

Nā, me whakaoti te whārite a=\frac{60±4\sqrt{1887}i}{48} ina he tango te ±. Tango 4i\sqrt{1887} mai i 60.

$$a=\frac{-4\sqrt{1887}i+60}{48}$$

Whakawehe 60-4i\sqrt{1887} ki te 48.

$$a=-\frac{\sqrt{1887}i}{12}+\frac{5}{4}$$

Kua oti te whārite te whakatau.

$$a=\frac{\sqrt{1887}i}{12}+\frac{5}{4} a=-\frac{\sqrt{1887}i}{12}+\frac{5}{4}$$

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.

$$24a^{2}-60a+352=0$$

Me tango 352 mai i ngā taha e rua o te whārite.

$$24a^{2}-60a+352-352=-352$$

Mā te tango i te 352 i a ia ake anō ka toe ko te 0.

$$24a^{2}-60a=-352$$

Whakawehea ngā taha e rua ki te 24.

$$\frac{24a^{2}-60a}{24}=-\frac{352}{24}$$

Mā te whakawehe ki te 24 ka wetekia te whakareanga ki te 24.

$$a^{2}+\left(-\frac{60}{24}\right)a=-\frac{352}{24}$$

Whakahekea te hautanga \frac{-60}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$a^{2}-\frac{5}{2}a=-\frac{352}{24}$$

Whakahekea te hautanga \frac{-352}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$a^{2}-\frac{5}{2}a=-\frac{44}{3}$$

Whakawehea te -\frac{5}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{4}. Nā, tāpiria te pūrua o te -\frac{5}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}-\frac{5}{2}a+\left(-\frac{5}{4}\right)^{2}=-\frac{44}{3}+\left(-\frac{5}{4}\right)^{2}$$

Pūruatia -\frac{5}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$a^{2}-\frac{5}{2}a+\frac{25}{16}=-\frac{44}{3}+\frac{25}{16}$$

Tāpiri -\frac{44}{3} ki te \frac{25}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a^{2}-\frac{5}{2}a+\frac{25}{16}=-\frac{629}{48}$$

Tauwehea a^{2}-\frac{5}{2}a+\frac{25}{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}.

$$\left(a-\frac{5}{4}\right)^{2}=-\frac{629}{48}$$

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

$$\sqrt{\left(a-\frac{5}{4}\right)^{2}}=\sqrt{-\frac{629}{48}}$$

Whakarūnātia.

$$a-\frac{5}{4}=\frac{\sqrt{1887}i}{12} a-\frac{5}{4}=-\frac{\sqrt{1887}i}{12}$$

Me tāpiri \frac{5}{4} ki ngā taha e rua o te whārite.

$$a=\frac{\sqrt{1887}i}{12}+\frac{5}{4} a=-\frac{\sqrt{1887}i}{12}+\frac{5}{4}$$