Whakaoti i te 24+5x-x^2=0

$$24 + 5 x - x ^ { 2 } = 0$$

Whakaoti mō x

$$x=-3$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+5x+24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=-24=-24$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -24.

$$-1,24 -2,12 -3,8 -4,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=8 b=-3$$

Tuhia anō te -x^{2}+5x+24 hei \left(-x^{2}+8x\right)+\left(-3x+24\right).

$$\left(-x^{2}+8x\right)+\left(-3x+24\right)$$

Tauwehea te -x i te tuatahi me te -3 i te rōpū tuarua.

$$-x\left(x-8\right)-3\left(x-8\right)$$

Whakatauwehea atu te kīanga pātahi x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-8\right)\left(-x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te -x-3=0.

$$x=8 x=-3$$

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.

$$-x^{2}+5x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 5 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-5±\sqrt{25+4\times 24}}{2\left(-1\right)}$$

Whakareatia 4 ki te 24.

$$x=\frac{-5±\sqrt{25+96}}{2\left(-1\right)}$$

Tāpiri 25 ki te 96.

$$x=\frac{-5±\sqrt{121}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-5±11}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-5±11}{-2}$$

Nā, me whakaoti te whārite x=\frac{-5±11}{-2} ina he tāpiri te ±. Tāpiri -5 ki te 11.

$$x=\frac{6}{-2}$$

Whakawehe 6 ki te -2.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{-5±11}{-2} ina he tango te ±. Tango 11 mai i -5.

$$x=-\frac{16}{-2}$$

Whakawehe -16 ki te -2.

$$x=8$$

Kua oti te whārite te whakatau.

$$x=-3 x=8$$

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.

$$-x^{2}+5x+24=0$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-x^{2}+5x+24-24=-24$$

Mā te tango i te 24 i a ia ake anō ka toe ko te 0.

$$-x^{2}+5x=-24$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+5x}{-1}=-\frac{24}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{5}{-1}x=-\frac{24}{-1}$$

Whakawehe 5 ki te -1.

$$x^{2}-5x=-\frac{24}{-1}$$

Whakawehe -24 ki te -1.

$$x^{2}-5x=24$$

Whakawehea te -5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{2}. Nā, tāpiria te pūrua o te -\frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=24+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=24+\frac{25}{4}$$

Tāpiri 24 ki te \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{121}{4}$$

Tauwehea x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{11}{2} x-\frac{5}{2}=-\frac{11}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=8 x=-3$$