Whakaoti i te x^2-9x-19/4=0

$${ x }^{ 2 } -9x- \frac{ 19 }{ 4 } = 0$$

Whakaoti mō x

$$x=-\frac{1}{2}=-0.5$$

$$x = \frac{19}{2} = 9\frac{1}{2} = 9.5$$

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.

$$x^{2}-9x-\frac{19}{4}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -9 mō b, me -\frac{19}{4} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-\frac{19}{4}\right)}}{2}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-\frac{19}{4}\right)}}{2}$$

Whakareatia -4 ki te -\frac{19}{4}.

$$x=\frac{-\left(-9\right)±\sqrt{81+19}}{2}$$

Tāpiri 81 ki te 19.

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

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-9\right)±10}{2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±10}{2}$$

Nā, me whakaoti te whārite x=\frac{9±10}{2} ina he tāpiri te ±. Tāpiri 9 ki te 10.

$$x=\frac{19}{2}$$

Nā, me whakaoti te whārite x=\frac{9±10}{2} ina he tango te ±. Tango 10 mai i 9.

$$x=-\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{19}{2} x=-\frac{1}{2}$$

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.

$$x^{2}-9x-\frac{19}{4}=0$$

Me tāpiri \frac{19}{4} ki ngā taha e rua o te whārite.

$$x^{2}-9x-\frac{19}{4}-\left(-\frac{19}{4}\right)=-\left(-\frac{19}{4}\right)$$

Mā te tango i te -\frac{19}{4} i a ia ake anō ka toe ko te 0.

$$x^{2}-9x=-\left(-\frac{19}{4}\right)$$

Tango -\frac{19}{4} mai i 0.

$$x^{2}-9x=\frac{19}{4}$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=\frac{19}{4}+\left(-\frac{9}{2}\right)^{2}$$

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

$$x^{2}-9x+\frac{81}{4}=\frac{19+81}{4}$$

Tāpiri \frac{19}{4} ki te \frac{81}{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.

$$x^{2}-9x+\frac{81}{4}=25$$

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

$$\left(x-\frac{9}{2}\right)^{2}=25$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

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

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=\frac{19}{2} x=-\frac{1}{2}$$