Whakaoti i te 10x^2+19x-15=0

$$10 { x }^{ 2 } +19x-15=0$$

Whakaoti mō x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x=\frac{3}{5}=0.6$$

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 10x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=19 ab=10\left(-15\right)=-150$$

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 -150.

$$-1,150 -2,75 -3,50 -5,30 -6,25 -10,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+150=149 -2+75=73 -3+50=47 -5+30=25 -6+25=19 -10+15=5$$

Ko te otinga te takirua ka hoatu i te tapeke 19.

$$a=-6 b=25$$

Tuhia anō te 10x^{2}+19x-15 hei \left(10x^{2}-6x\right)+\left(25x-15\right).

$$\left(10x^{2}-6x\right)+\left(25x-15\right)$$

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

$$2x\left(5x-3\right)+5\left(5x-3\right)$$

Whakatauwehea atu te kīanga pātahi 5x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x-3\right)\left(2x+5\right)$$

Hei kimi otinga whārite, me whakaoti te 5x-3=0 me te 2x+5=0.

$$x=\frac{3}{5} x=-\frac{5}{2}$$

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.

$$10x^{2}+19x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, 19 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\times 10\left(-15\right)}}{2\times 10}$$

Pūrua 19.

$$x=\frac{-19±\sqrt{361-4\times 10\left(-15\right)}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-19±\sqrt{361-40\left(-15\right)}}{2\times 10}$$

Whakareatia -40 ki te -15.

$$x=\frac{-19±\sqrt{361+600}}{2\times 10}$$

Tāpiri 361 ki te 600.

$$x=\frac{-19±\sqrt{961}}{2\times 10}$$

Tuhia te pūtakerua o te 961.

$$x=\frac{-19±31}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{-19±31}{20}$$

Nā, me whakaoti te whārite x=\frac{-19±31}{20} ina he tāpiri te ±. Tāpiri -19 ki te 31.

$$x=\frac{12}{20}$$

Whakahekea te hautanga \frac{12}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{3}{5}$$

Nā, me whakaoti te whārite x=\frac{-19±31}{20} ina he tango te ±. Tango 31 mai i -19.

$$x=-\frac{50}{20}$$

Whakahekea te hautanga \frac{-50}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

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

Kua oti te whārite te whakatau.

$$x=\frac{3}{5} x=-\frac{5}{2}$$

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.

$$10x^{2}+19x-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$10x^{2}+19x-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$10x^{2}+19x=-\left(-15\right)$$

Tango -15 mai i 0.

$$10x^{2}+19x=15$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10x^{2}+19x}{10}=\frac{15}{10}$$

Mā te whakawehe ki te 10 ka wetekia te whakareanga ki te 10.

$$x^{2}+\frac{19}{10}x=\frac{15}{10}$$

Whakahekea te hautanga \frac{15}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{19}{10}x=\frac{3}{2}$$

Whakawehea te \frac{19}{10}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{19}{20}. Nā, tāpiria te pūrua o te \frac{19}{20} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{19}{10}x+\left(\frac{19}{20}\right)^{2}=\frac{3}{2}+\left(\frac{19}{20}\right)^{2}$$

Pūruatia \frac{19}{20} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{19}{10}x+\frac{361}{400}=\frac{3}{2}+\frac{361}{400}$$

Tāpiri \frac{3}{2} ki te \frac{361}{400} 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.

$$x^{2}+\frac{19}{10}x+\frac{361}{400}=\frac{961}{400}$$

Tauwehea te x^{2}+\frac{19}{10}x+\frac{361}{400}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{19}{20}\right)^{2}=\frac{961}{400}$$

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

$$\sqrt{\left(x+\frac{19}{20}\right)^{2}}=\sqrt{\frac{961}{400}}$$

Whakarūnātia.

$$x+\frac{19}{20}=\frac{31}{20} x+\frac{19}{20}=-\frac{31}{20}$$

Me tango \frac{19}{20} mai i ngā taha e rua o te whārite.

$$x=\frac{3}{5} x=-\frac{5}{2}$$