Whakaoti i te 2x^2+5x=8

$$2 x ^ { 2 } + 5 x = 8$$

Whakaoti mō x

$$x = \frac{\sqrt{89} - 5}{4} \approx 1.108495283$$

$$x=\frac{-\sqrt{89}-5}{4}\approx -3.608495283$$

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.

$$2x^{2}+5x=8$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$2x^{2}+5x-8=8-8$$

Mā te tango i te 8 i a ia ake anō ka toe ko te 0.

$$2x^{2}+5x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 5 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-8\right)}}{2\times 2}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-8\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-5±\sqrt{25-8\left(-8\right)}}{2\times 2}$$

Whakareatia -8 ki te -8.

$$x=\frac{-5±\sqrt{25+64}}{2\times 2}$$

Tāpiri 25 ki te 64.

$$x=\frac{-5±\sqrt{89}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-5±\sqrt{89}}{4}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{89}}{4} ina he tāpiri te ±. Tāpiri -5 ki te \sqrt{89}.

$$x=\frac{\sqrt{89}-5}{4}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{89}}{4} ina he tango te ±. Tango \sqrt{89} mai i -5.

$$x=\frac{-\sqrt{89}-5}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{89}-5}{4} x=\frac{-\sqrt{89}-5}{4}$$

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.

$$2x^{2}+5x=8$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+5x}{2}=\frac{8}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\frac{5}{2}x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x^{2}+\frac{5}{2}x=4$$

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.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=4+\left(\frac{5}{4}\right)^{2}$$

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

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=4+\frac{25}{16}$$

Tāpiri 4 ki te \frac{25}{16}.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{89}{16}$$

Tauwehea te x^{2}+\frac{5}{2}x+\frac{25}{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}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{89}{16}$$

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

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{89}{16}}$$

Whakarūnātia.

$$x+\frac{5}{4}=\frac{\sqrt{89}}{4} x+\frac{5}{4}=-\frac{\sqrt{89}}{4}$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{89}-5}{4} x=\frac{-\sqrt{89}-5}{4}$$