Whakaoti i te -3x^2+51x-156=0

$$- 3 x ^ { 2 } + 51 x - 156 = 0$$

Whakaoti mō x

$$x=4$$

$$x=13$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 3.

$$-x^{2}+17x-52=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-52. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=17 ab=-\left(-52\right)=52$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 52.

$$1,52 2,26 4,13$$

Tātaihia te tapeke mō ia takirua.

$$1+52=53 2+26=28 4+13=17$$

Ko te otinga te takirua ka hoatu i te tapeke 17.

$$a=13 b=4$$

Tuhia anō te -x^{2}+17x-52 hei \left(-x^{2}+13x\right)+\left(4x-52\right).

$$\left(-x^{2}+13x\right)+\left(4x-52\right)$$

Tauwehea te -x i te tuatahi me te 4 i te rōpū tuarua.

$$-x\left(x-13\right)+4\left(x-13\right)$$

Whakatauwehea atu te kīanga pātahi x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-13\right)\left(-x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te -x+4=0.

$$x=13 x=4$$

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.

$$-3x^{2}+51x-156=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 51 mō b, me -156 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-51±\sqrt{51^{2}-4\left(-3\right)\left(-156\right)}}{2\left(-3\right)}$$

Pūrua 51.

$$x=\frac{-51±\sqrt{2601-4\left(-3\right)\left(-156\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-51±\sqrt{2601+12\left(-156\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -156.

$$x=\frac{-51±\sqrt{2601-1872}}{2\left(-3\right)}$$

Tāpiri 2601 ki te -1872.

$$x=\frac{-51±\sqrt{729}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-51±27}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-51±27}{-6}$$

Nā, me whakaoti te whārite x=\frac{-51±27}{-6} ina he tāpiri te ±. Tāpiri -51 ki te 27.

$$x=-\frac{24}{-6}$$

Whakawehe -24 ki te -6.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-51±27}{-6} ina he tango te ±. Tango 27 mai i -51.

$$x=-\frac{78}{-6}$$

Whakawehe -78 ki te -6.

$$x=13$$

Kua oti te whārite te whakatau.

$$x=4 x=13$$

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.

$$-3x^{2}+51x-156=0$$

Me tāpiri 156 ki ngā taha e rua o te whārite.

$$-3x^{2}+51x-156-\left(-156\right)=-\left(-156\right)$$

Mā te tango i te -156 i a ia ake anō ka toe ko te 0.

$$-3x^{2}+51x=-\left(-156\right)$$

Tango -156 mai i 0.

$$-3x^{2}+51x=156$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+51x}{-3}=\frac{156}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\frac{51}{-3}x=\frac{156}{-3}$$

Whakawehe 51 ki te -3.

$$x^{2}-17x=\frac{156}{-3}$$

Whakawehe 156 ki te -3.

$$x^{2}-17x=-52$$

Whakawehea te -17, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{17}{2}. Nā, tāpiria te pūrua o te -\frac{17}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-52+\left(-\frac{17}{2}\right)^{2}$$

Pūruatia -\frac{17}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-17x+\frac{289}{4}=-52+\frac{289}{4}$$

Tāpiri -52 ki te \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{81}{4}$$

Tauwehea x^{2}-17x+\frac{289}{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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{81}{4}$$

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

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Whakarūnātia.

$$x-\frac{17}{2}=\frac{9}{2} x-\frac{17}{2}=-\frac{9}{2}$$

Me tāpiri \frac{17}{2} ki ngā taha e rua o te whārite.

$$x=13 x=4$$