Whakaoti i te -4b^2+22b-4=0

$$- 4 b ^ { 2 } + 22 b - 4 = 0$$

Whakaoti mō b

$$b = \frac{\sqrt{105} + 11}{4} \approx 5.311737691$$

$$b=\frac{11-\sqrt{105}}{4}\approx 0.188262309$$

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.

$$-4b^{2}+22b-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, 22 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-22±\sqrt{22^{2}-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

Pūrua 22.

$$b=\frac{-22±\sqrt{484-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$b=\frac{-22±\sqrt{484+16\left(-4\right)}}{2\left(-4\right)}$$

Whakareatia 16 ki te -4.

$$b=\frac{-22±\sqrt{484-64}}{2\left(-4\right)}$$

Tāpiri 484 ki te -64.

$$b=\frac{-22±\sqrt{420}}{2\left(-4\right)}$$

Tuhia te pūtakerua o te 420.

$$b=\frac{-22±2\sqrt{105}}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$b=\frac{-22±2\sqrt{105}}{-8}$$

Nā, me whakaoti te whārite b=\frac{-22±2\sqrt{105}}{-8} ina he tāpiri te ±. Tāpiri -22 ki te 2\sqrt{105}.

$$b=\frac{2\sqrt{105}-22}{-8}$$

Whakawehe -22+2\sqrt{105} ki te -8.

$$b=\frac{11-\sqrt{105}}{4}$$

Nā, me whakaoti te whārite b=\frac{-22±2\sqrt{105}}{-8} ina he tango te ±. Tango 2\sqrt{105} mai i -22.

$$b=\frac{-2\sqrt{105}-22}{-8}$$

Whakawehe -22-2\sqrt{105} ki te -8.

$$b=\frac{\sqrt{105}+11}{4}$$

Kua oti te whārite te whakatau.

$$b=\frac{11-\sqrt{105}}{4} b=\frac{\sqrt{105}+11}{4}$$

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.

$$-4b^{2}+22b-4=0$$

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

$$-4b^{2}+22b-4-\left(-4\right)=-\left(-4\right)$$

Mā te tango i te -4 i a ia ake anō ka toe ko te 0.

$$-4b^{2}+22b=-\left(-4\right)$$

Tango -4 mai i 0.

$$-4b^{2}+22b=4$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4b^{2}+22b}{-4}=\frac{4}{-4}$$

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

$$b^{2}+\frac{22}{-4}b=\frac{4}{-4}$$

Whakahekea te hautanga \frac{22}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$b^{2}-\frac{11}{2}b=\frac{4}{-4}$$

Whakawehe 4 ki te -4.

$$b^{2}-\frac{11}{2}b=-1$$

Whakawehea te -\frac{11}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{4}. Nā, tāpiria te pūrua o te -\frac{11}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$b^{2}-\frac{11}{2}b+\left(-\frac{11}{4}\right)^{2}=-1+\left(-\frac{11}{4}\right)^{2}$$

Pūruatia -\frac{11}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$b^{2}-\frac{11}{2}b+\frac{121}{16}=-1+\frac{121}{16}$$

Tāpiri -1 ki te \frac{121}{16}.

$$b^{2}-\frac{11}{2}b+\frac{121}{16}=\frac{105}{16}$$

Tauwehea te b^{2}-\frac{11}{2}b+\frac{121}{16}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-\frac{11}{4}\right)^{2}=\frac{105}{16}$$

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

$$\sqrt{\left(b-\frac{11}{4}\right)^{2}}=\sqrt{\frac{105}{16}}$$

Whakarūnātia.

$$b-\frac{11}{4}=\frac{\sqrt{105}}{4} b-\frac{11}{4}=-\frac{\sqrt{105}}{4}$$

Me tāpiri \frac{11}{4} ki ngā taha e rua o te whārite.

$$b=\frac{\sqrt{105}+11}{4} b=\frac{11-\sqrt{105}}{4}$$