Whakaoti i te 4x^2+15x-25=0

$$4 x ^ { 2 } + 15 x - 25 = 0$$

Whakaoti mō x

$$x=-5$$

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx-25. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=15 ab=4\left(-25\right)=-100$$

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 -100.

$$-1,100 -2,50 -4,25 -5,20 -10,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+100=99 -2+50=48 -4+25=21 -5+20=15 -10+10=0$$

Ko te otinga te takirua ka hoatu i te tapeke 15.

$$a=-5 b=20$$

Tuhia anō te 4x^{2}+15x-25 hei \left(4x^{2}-5x\right)+\left(20x-25\right).

$$\left(4x^{2}-5x\right)+\left(20x-25\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(4x-5\right)+5\left(4x-5\right)$$

Whakatauwehea atu te kīanga pātahi 4x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-5\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te 4x-5=0 me te x+5=0.

$$x=\frac{5}{4} x=-5$$

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.

$$4x^{2}+15x-25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 15 mō b, me -25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 4\left(-25\right)}}{2\times 4}$$

Pūrua 15.

$$x=\frac{-15±\sqrt{225-4\times 4\left(-25\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-15±\sqrt{225-16\left(-25\right)}}{2\times 4}$$

Whakareatia -16 ki te -25.

$$x=\frac{-15±\sqrt{225+400}}{2\times 4}$$

Tāpiri 225 ki te 400.

$$x=\frac{-15±\sqrt{625}}{2\times 4}$$

Tuhia te pūtakerua o te 625.

$$x=\frac{-15±25}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-15±25}{8}$$

Nā, me whakaoti te whārite x=\frac{-15±25}{8} ina he tāpiri te ±. Tāpiri -15 ki te 25.

$$x=\frac{10}{8}$$

Whakahekea te hautanga \frac{10}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{5}{4}$$

Nā, me whakaoti te whārite x=\frac{-15±25}{8} ina he tango te ±. Tango 25 mai i -15.

$$x=-\frac{40}{8}$$

Whakawehe -40 ki te 8.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=\frac{5}{4} x=-5$$

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.

$$4x^{2}+15x-25=0$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$4x^{2}+15x-25-\left(-25\right)=-\left(-25\right)$$

Mā te tango i te -25 i a ia ake anō ka toe ko te 0.

$$4x^{2}+15x=-\left(-25\right)$$

Tango -25 mai i 0.

$$4x^{2}+15x=25$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+15x}{4}=\frac{25}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\frac{15}{4}x=\frac{25}{4}$$

Whakawehea te \frac{15}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{8}. Nā, tāpiria te pūrua o te \frac{15}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{15}{4}x+\left(\frac{15}{8}\right)^{2}=\frac{25}{4}+\left(\frac{15}{8}\right)^{2}$$

Pūruatia \frac{15}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{15}{4}x+\frac{225}{64}=\frac{25}{4}+\frac{225}{64}$$

Tāpiri \frac{25}{4} ki te \frac{225}{64} 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.

$$x^{2}+\frac{15}{4}x+\frac{225}{64}=\frac{625}{64}$$

Tauwehea x^{2}+\frac{15}{4}x+\frac{225}{64}. 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}.

$$\left(x+\frac{15}{8}\right)^{2}=\frac{625}{64}$$

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

$$\sqrt{\left(x+\frac{15}{8}\right)^{2}}=\sqrt{\frac{625}{64}}$$

Whakarūnātia.

$$x+\frac{15}{8}=\frac{25}{8} x+\frac{15}{8}=-\frac{25}{8}$$

Me tango \frac{15}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{5}{4} x=-5$$