Whakaoti i te 4x^2-7x-9=0

$$4 x ^ { 2 } - 7 x - 9 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{193} + 7}{8} \approx 2.611555499$$

$$x=\frac{7-\sqrt{193}}{8}\approx -0.861555499$$

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.

$$4x^{2}-7x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -7 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 4\left(-9\right)}}{2\times 4}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 4\left(-9\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-7\right)±\sqrt{49-16\left(-9\right)}}{2\times 4}$$

Whakareatia -16 ki te -9.

$$x=\frac{-\left(-7\right)±\sqrt{49+144}}{2\times 4}$$

Tāpiri 49 ki te 144.

$$x=\frac{-\left(-7\right)±\sqrt{193}}{2\times 4}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±\sqrt{193}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{7±\sqrt{193}}{8}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{193}}{8} ina he tāpiri te ±. Tāpiri 7 ki te \sqrt{193}.

$$x=\frac{\sqrt{193}+7}{8}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{193}}{8} ina he tango te ±. Tango \sqrt{193} mai i 7.

$$x=\frac{7-\sqrt{193}}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{193}+7}{8} x=\frac{7-\sqrt{193}}{8}$$

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.

$$4x^{2}-7x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$4x^{2}-7x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$4x^{2}-7x=-\left(-9\right)$$

Tango -9 mai i 0.

$$4x^{2}-7x=9$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-7x}{4}=\frac{9}{4}$$

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

$$x^{2}-\frac{7}{4}x=\frac{9}{4}$$

Whakawehea te -\frac{7}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{8}. Nā, tāpiria te pūrua o te -\frac{7}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{4}x+\left(-\frac{7}{8}\right)^{2}=\frac{9}{4}+\left(-\frac{7}{8}\right)^{2}$$

Pūruatia -\frac{7}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{9}{4}+\frac{49}{64}$$

Tāpiri \frac{9}{4} ki te \frac{49}{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.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{193}{64}$$

Tauwehea x^{2}-\frac{7}{4}x+\frac{49}{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}.

$$\left(x-\frac{7}{8}\right)^{2}=\frac{193}{64}$$

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

$$\sqrt{\left(x-\frac{7}{8}\right)^{2}}=\sqrt{\frac{193}{64}}$$

Whakarūnātia.

$$x-\frac{7}{8}=\frac{\sqrt{193}}{8} x-\frac{7}{8}=-\frac{\sqrt{193}}{8}$$

Me tāpiri \frac{7}{8} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{193}+7}{8} x=\frac{7-\sqrt{193}}{8}$$