Whakaoti i te -4a^2-5a+1=0

$$- 4 a ^ { 2 } - 5 a + 1 = 0$$

Whakaoti mō a

$$a=\frac{\sqrt{41}-5}{8}\approx 0.17539053$$

$$a=\frac{-\sqrt{41}-5}{8}\approx -1.42539053$$

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.

$$-4a^{2}-5a+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, -5 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -5.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\left(-4\right)}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$a=\frac{-\left(-5\right)±\sqrt{25+16}}{2\left(-4\right)}$$

Tāpiri 25 ki te 16.

$$a=\frac{-\left(-5\right)±\sqrt{41}}{2\left(-4\right)}$$

Ko te tauaro o -5 ko 5.

$$a=\frac{5±\sqrt{41}}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$a=\frac{5±\sqrt{41}}{-8}$$

Nā, me whakaoti te whārite a=\frac{5±\sqrt{41}}{-8} ina he tāpiri te ±. Tāpiri 5 ki te \sqrt{41}.

$$a=\frac{\sqrt{41}+5}{-8}$$

Whakawehe 5+\sqrt{41} ki te -8.

$$a=\frac{-\sqrt{41}-5}{8}$$

Nā, me whakaoti te whārite a=\frac{5±\sqrt{41}}{-8} ina he tango te ±. Tango \sqrt{41} mai i 5.

$$a=\frac{5-\sqrt{41}}{-8}$$

Whakawehe 5-\sqrt{41} ki te -8.

$$a=\frac{\sqrt{41}-5}{8}$$

Kua oti te whārite te whakatau.

$$a=\frac{-\sqrt{41}-5}{8} a=\frac{\sqrt{41}-5}{8}$$

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.

$$-4a^{2}-5a+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-4a^{2}-5a+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$-4a^{2}-5a=-1$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4a^{2}-5a}{-4}=-\frac{1}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$a^{2}+\left(-\frac{5}{-4}\right)a=-\frac{1}{-4}$$

Whakawehe -5 ki te -4.

$$a^{2}+\frac{5}{4}a=-\frac{1}{-4}$$

Whakawehe -1 ki te -4.

$$a^{2}+\frac{5}{4}a=\frac{1}{4}$$

Whakawehea te \frac{5}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{8}. Nā, tāpiria te pūrua o te \frac{5}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}+\frac{5}{4}a+\left(\frac{5}{8}\right)^{2}=\frac{1}{4}+\left(\frac{5}{8}\right)^{2}$$

Pūruatia \frac{5}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$a^{2}+\frac{5}{4}a+\frac{25}{64}=\frac{1}{4}+\frac{25}{64}$$

Tāpiri \frac{1}{4} ki te \frac{25}{64} 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.

$$a^{2}+\frac{5}{4}a+\frac{25}{64}=\frac{41}{64}$$

Tauwehea a^{2}+\frac{5}{4}a+\frac{25}{64}. 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}.

$$\left(a+\frac{5}{8}\right)^{2}=\frac{41}{64}$$

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

$$\sqrt{\left(a+\frac{5}{8}\right)^{2}}=\sqrt{\frac{41}{64}}$$

Whakarūnātia.

$$a+\frac{5}{8}=\frac{\sqrt{41}}{8} a+\frac{5}{8}=-\frac{\sqrt{41}}{8}$$

Me tango \frac{5}{8} mai i ngā taha e rua o te whārite.

$$a=\frac{\sqrt{41}-5}{8} a=\frac{-\sqrt{41}-5}{8}$$