Whakaoti i te 7x-5/2x^2-5/2x=1000

$$7x- \frac{ 5 }{ 2 } { x }^{ 2 } - \frac{ 5 }{ 2 } x=1000$$

Whakaoti mō x (complex solution)

$$x=\frac{9+\sqrt{39919}i}{10}\approx 0.9+19.979739738i$$

$$x=\frac{-\sqrt{39919}i+9}{10}\approx 0.9-19.979739738i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te 7x me -\frac{5}{2}x, ka \frac{9}{2}x.

$$\frac{9}{2}x-\frac{5}{2}x^{2}=1000$$

Tangohia te 1000 mai i ngā taha e rua.

$$\frac{9}{2}x-\frac{5}{2}x^{2}-1000=0$$

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.

$$-\frac{5}{2}x^{2}+\frac{9}{2}x-1000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{5}{2} mō a, \frac{9}{2} mō b, me -1000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{2}±\sqrt{\left(\frac{9}{2}\right)^{2}-4\left(-\frac{5}{2}\right)\left(-1000\right)}}{2\left(-\frac{5}{2}\right)}$$

Pūruatia \frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-4\left(-\frac{5}{2}\right)\left(-1000\right)}}{2\left(-\frac{5}{2}\right)}$$

Whakareatia -4 ki te -\frac{5}{2}.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}+10\left(-1000\right)}}{2\left(-\frac{5}{2}\right)}$$

Whakareatia 10 ki te -1000.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-10000}}{2\left(-\frac{5}{2}\right)}$$

Tāpiri \frac{81}{4} ki te -10000.

$$x=\frac{-\frac{9}{2}±\sqrt{-\frac{39919}{4}}}{2\left(-\frac{5}{2}\right)}$$

Tuhia te pūtakerua o te -\frac{39919}{4}.

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{2\left(-\frac{5}{2}\right)}$$

Whakareatia 2 ki te -\frac{5}{2}.

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5}$$

Nā, me whakaoti te whārite x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5} ina he tāpiri te ±. Tāpiri -\frac{9}{2} ki te \frac{i\sqrt{39919}}{2}.

$$x=\frac{-9+\sqrt{39919}i}{-5\times 2}$$

Whakawehe \frac{-9+i\sqrt{39919}}{2} ki te -5.

$$x=\frac{-\sqrt{39919}i+9}{10}$$

Nā, me whakaoti te whārite x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5} ina he tango te ±. Tango \frac{i\sqrt{39919}}{2} mai i -\frac{9}{2}.

$$x=\frac{-\sqrt{39919}i-9}{-5\times 2}$$

Whakawehe \frac{-9-i\sqrt{39919}}{2} ki te -5.

$$x=\frac{9+\sqrt{39919}i}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{39919}i+9}{10} x=\frac{9+\sqrt{39919}i}{10}$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te 7x me -\frac{5}{2}x, ka \frac{9}{2}x.

$$\frac{9}{2}x-\frac{5}{2}x^{2}=1000$$

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.

$$-\frac{5}{2}x^{2}+\frac{9}{2}x=1000$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-\frac{5}{2}x^{2}+\frac{9}{2}x}{-\frac{5}{2}}=\frac{1000}{-\frac{5}{2}}$$

Mā te whakawehe ki te -\frac{5}{2} ka wetekia te whakareanga ki te -\frac{5}{2}.

$$x^{2}+\frac{\frac{9}{2}}{-\frac{5}{2}}x=\frac{1000}{-\frac{5}{2}}$$

Whakawehe \frac{9}{2} ki te -\frac{5}{2} mā te whakarea \frac{9}{2} ki te tau huripoki o -\frac{5}{2}.

$$x^{2}-\frac{9}{5}x=\frac{1000}{-\frac{5}{2}}$$

Whakawehe 1000 ki te -\frac{5}{2} mā te whakarea 1000 ki te tau huripoki o -\frac{5}{2}.

$$x^{2}-\frac{9}{5}x=-400$$

Whakawehea te -\frac{9}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{10}. Nā, tāpiria te pūrua o te -\frac{9}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{5}x+\left(-\frac{9}{10}\right)^{2}=-400+\left(-\frac{9}{10}\right)^{2}$$

Pūruatia -\frac{9}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=-400+\frac{81}{100}$$

Tāpiri -400 ki te \frac{81}{100}.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=-\frac{39919}{100}$$

Tauwehea te x^{2}-\frac{9}{5}x+\frac{81}{100}. 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}.

$$\left(x-\frac{9}{10}\right)^{2}=-\frac{39919}{100}$$

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

$$\sqrt{\left(x-\frac{9}{10}\right)^{2}}=\sqrt{-\frac{39919}{100}}$$

Whakarūnātia.

$$x-\frac{9}{10}=\frac{\sqrt{39919}i}{10} x-\frac{9}{10}=-\frac{\sqrt{39919}i}{10}$$

Me tāpiri \frac{9}{10} ki ngā taha e rua o te whārite.

$$x=\frac{9+\sqrt{39919}i}{10} x=\frac{-\sqrt{39919}i+9}{10}$$