Whakaoti i te x^2-7x+3=0

$$x ^ { 2 } - 7 x + 3 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{37} + 7}{2} \approx 6.541381265$$

$$x=\frac{7-\sqrt{37}}{2}\approx 0.458618735$$

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.

$$x^{2}-7x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -7 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3}}{2}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3}}{2}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-12}}{2}$$

Tāpiri 49 ki te -12.

$$x=\frac{-\left(-7\right)±\sqrt{37}}{2}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±\sqrt{37}}{2}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{37}}{2} ina he tāpiri te ±. Tāpiri 7 ki te \sqrt{37}.

$$x=\frac{\sqrt{37}+7}{2}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{37}}{2} ina he tango te ±. Tango \sqrt{37} mai i 7.

$$x=\frac{7-\sqrt{37}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{37}+7}{2} x=\frac{7-\sqrt{37}}{2}$$

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.

$$x^{2}-7x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x^{2}-7x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$x^{2}-7x=-3$$

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.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-3+\left(-\frac{7}{2}\right)^{2}$$

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

$$x^{2}-7x+\frac{49}{4}=-3+\frac{49}{4}$$

Tāpiri -3 ki te \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{37}{4}$$

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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{37}{4}$$

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

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Whakarūnātia.

$$x-\frac{7}{2}=\frac{\sqrt{37}}{2} x-\frac{7}{2}=-\frac{\sqrt{37}}{2}$$

Me tāpiri \frac{7}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{37}+7}{2} x=\frac{7-\sqrt{37}}{2}$$