Whakaoti i te 4x^2+28x+53=0

$$4 x ^ { 2 } + 28 x + 53 = 0$$

Whakaoti mō x (complex solution)

$$x=-\frac{7}{2}+i=-3.5+i$$

$$x=-\frac{7}{2}-i=-3.5-i$$

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.

$$4x^{2}+28x+53=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 28 mō b, me 53 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 4\times 53}}{2\times 4}$$

Pūrua 28.

$$x=\frac{-28±\sqrt{784-4\times 4\times 53}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-28±\sqrt{784-16\times 53}}{2\times 4}$$

Whakareatia -16 ki te 53.

$$x=\frac{-28±\sqrt{784-848}}{2\times 4}$$

Tāpiri 784 ki te -848.

$$x=\frac{-28±\sqrt{-64}}{2\times 4}$$

Tuhia te pūtakerua o te -64.

$$x=\frac{-28±8i}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-28±8i}{8}$$

Nā, me whakaoti te whārite x=\frac{-28±8i}{8} ina he tāpiri te ±. Tāpiri -28 ki te 8i.

$$x=\frac{-28+8i}{8}$$

Whakawehe -28+8i ki te 8.

$$x=-\frac{7}{2}+i$$

Nā, me whakaoti te whārite x=\frac{-28±8i}{8} ina he tango te ±. Tango 8i mai i -28.

$$x=\frac{-28-8i}{8}$$

Whakawehe -28-8i ki te 8.

$$x=-\frac{7}{2}-i$$

Kua oti te whārite te whakatau.

$$x=-\frac{7}{2}+i x=-\frac{7}{2}-i$$

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.

$$4x^{2}+28x+53=0$$

Me tango 53 mai i ngā taha e rua o te whārite.

$$4x^{2}+28x+53-53=-53$$

Mā te tango i te 53 i a ia ake anō ka toe ko te 0.

$$4x^{2}+28x=-53$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+28x}{4}=-\frac{53}{4}$$

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

$$x^{2}+\frac{28}{4}x=-\frac{53}{4}$$

Whakawehe 28 ki te 4.

$$x^{2}+7x=-\frac{53}{4}$$

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

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-\frac{53}{4}+\left(\frac{7}{2}\right)^{2}$$

Pūruatia \frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+7x+\frac{49}{4}=\frac{-53+49}{4}$$

Tāpiri -\frac{53}{4} ki te \frac{49}{4} 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.

$$x^{2}+7x+\frac{49}{4}=-1$$

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

$$\left(x+\frac{7}{2}\right)^{2}=-1$$

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

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{-1}$$

Whakarūnātia.

$$x+\frac{7}{2}=i x+\frac{7}{2}=-i$$

Me tango \frac{7}{2} mai i ngā taha e rua o te whārite.

$$x=-\frac{7}{2}+i x=-\frac{7}{2}-i$$