Whakaoti i te x^2-489x+28980=0

$$x ^ { 2 } - 489 x + 28980 = 0$$

Whakaoti mō x

$$x=69$$

$$x=420$$

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.

$$x^{2}-489x+28980=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -489 mō b, me 28980 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-489\right)±\sqrt{\left(-489\right)^{2}-4\times 28980}}{2}$$

Pūrua -489.

$$x=\frac{-\left(-489\right)±\sqrt{239121-4\times 28980}}{2}$$

Whakareatia -4 ki te 28980.

$$x=\frac{-\left(-489\right)±\sqrt{239121-115920}}{2}$$

Tāpiri 239121 ki te -115920.

$$x=\frac{-\left(-489\right)±\sqrt{123201}}{2}$$

Tuhia te pūtakerua o te 123201.

$$x=\frac{-\left(-489\right)±351}{2}$$

Ko te tauaro o -489 ko 489.

$$x=\frac{489±351}{2}$$

Nā, me whakaoti te whārite x=\frac{489±351}{2} ina he tāpiri te ±. Tāpiri 489 ki te 351.

$$x=\frac{840}{2}$$

Whakawehe 840 ki te 2.

$$x=420$$

Nā, me whakaoti te whārite x=\frac{489±351}{2} ina he tango te ±. Tango 351 mai i 489.

$$x=\frac{138}{2}$$

Whakawehe 138 ki te 2.

$$x=69$$

Kua oti te whārite te whakatau.

$$x=420 x=69$$

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.

$$x^{2}-489x+28980=0$$

Me tango 28980 mai i ngā taha e rua o te whārite.

$$x^{2}-489x+28980-28980=-28980$$

Mā te tango i te 28980 i a ia ake anō ka toe ko te 0.

$$x^{2}-489x=-28980$$

Whakawehea te -489, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{489}{2}. Nā, tāpiria te pūrua o te -\frac{489}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-489x+\left(-\frac{489}{2}\right)^{2}=-28980+\left(-\frac{489}{2}\right)^{2}$$

Pūruatia -\frac{489}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-489x+\frac{239121}{4}=-28980+\frac{239121}{4}$$

Tāpiri -28980 ki te \frac{239121}{4}.

$$x^{2}-489x+\frac{239121}{4}=\frac{123201}{4}$$

Tauwehea x^{2}-489x+\frac{239121}{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}.

$$\left(x-\frac{489}{2}\right)^{2}=\frac{123201}{4}$$

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

$$\sqrt{\left(x-\frac{489}{2}\right)^{2}}=\sqrt{\frac{123201}{4}}$$

Whakarūnātia.

$$x-\frac{489}{2}=\frac{351}{2} x-\frac{489}{2}=-\frac{351}{2}$$

Me tāpiri \frac{489}{2} ki ngā taha e rua o te whārite.

$$x=420 x=69$$