Whakaoti i te 49x^2+30x+25=0

$$49 x ^ { 2 } + 30 x + 25 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-15+10\sqrt{10}i}{49}\approx -0.306122449+0.645362788i$$

$$x=\frac{-10\sqrt{10}i-15}{49}\approx -0.306122449-0.645362788i$$

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.

$$49x^{2}+30x+25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 49 mō a, 30 mō b, me 25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 49\times 25}}{2\times 49}$$

Pūrua 30.

$$x=\frac{-30±\sqrt{900-4\times 49\times 25}}{2\times 49}$$

Whakareatia -4 ki te 49.

$$x=\frac{-30±\sqrt{900-196\times 25}}{2\times 49}$$

Whakareatia -196 ki te 25.

$$x=\frac{-30±\sqrt{900-4900}}{2\times 49}$$

Tāpiri 900 ki te -4900.

$$x=\frac{-30±\sqrt{-4000}}{2\times 49}$$

Tuhia te pūtakerua o te -4000.

$$x=\frac{-30±20\sqrt{10}i}{2\times 49}$$

Whakareatia 2 ki te 49.

$$x=\frac{-30±20\sqrt{10}i}{98}$$

Nā, me whakaoti te whārite x=\frac{-30±20\sqrt{10}i}{98} ina he tāpiri te ±. Tāpiri -30 ki te 20i\sqrt{10}.

$$x=\frac{-30+20\sqrt{10}i}{98}$$

Whakawehe -30+20i\sqrt{10} ki te 98.

$$x=\frac{-15+10\sqrt{10}i}{49}$$

Nā, me whakaoti te whārite x=\frac{-30±20\sqrt{10}i}{98} ina he tango te ±. Tango 20i\sqrt{10} mai i -30.

$$x=\frac{-20\sqrt{10}i-30}{98}$$

Whakawehe -30-20i\sqrt{10} ki te 98.

$$x=\frac{-10\sqrt{10}i-15}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{-15+10\sqrt{10}i}{49} x=\frac{-10\sqrt{10}i-15}{49}$$

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.

$$49x^{2}+30x+25=0$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$49x^{2}+30x+25-25=-25$$

Mā te tango i te 25 i a ia ake anō ka toe ko te 0.

$$49x^{2}+30x=-25$$

Whakawehea ngā taha e rua ki te 49.

$$\frac{49x^{2}+30x}{49}=-\frac{25}{49}$$

Mā te whakawehe ki te 49 ka wetekia te whakareanga ki te 49.

$$x^{2}+\frac{30}{49}x=-\frac{25}{49}$$

Whakawehea te \frac{30}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{49}. Nā, tāpiria te pūrua o te \frac{15}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{30}{49}x+\left(\frac{15}{49}\right)^{2}=-\frac{25}{49}+\left(\frac{15}{49}\right)^{2}$$

Pūruatia \frac{15}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{30}{49}x+\frac{225}{2401}=-\frac{25}{49}+\frac{225}{2401}$$

Tāpiri -\frac{25}{49} ki te \frac{225}{2401} 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.

$$x^{2}+\frac{30}{49}x+\frac{225}{2401}=-\frac{1000}{2401}$$

Tauwehea x^{2}+\frac{30}{49}x+\frac{225}{2401}. 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}.

$$\left(x+\frac{15}{49}\right)^{2}=-\frac{1000}{2401}$$

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

$$\sqrt{\left(x+\frac{15}{49}\right)^{2}}=\sqrt{-\frac{1000}{2401}}$$

Whakarūnātia.

$$x+\frac{15}{49}=\frac{10\sqrt{10}i}{49} x+\frac{15}{49}=-\frac{10\sqrt{10}i}{49}$$

Me tango \frac{15}{49} mai i ngā taha e rua o te whārite.

$$x=\frac{-15+10\sqrt{10}i}{49} x=\frac{-10\sqrt{10}i-15}{49}$$