Whakaoti i te 30x+21x^2=3384

$$30 x + 21 x ^ { 2 } = 3384$$

Whakaoti mō x

$$x = -\frac{94}{7} = -13\frac{3}{7} \approx -13.428571429$$

$$x=12$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3384 mai i ngā taha e rua.

$$30x+21x^{2}-3384=0$$

Whakawehea ngā taha e rua ki te 3.

$$10x+7x^{2}-1128=0$$

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

$$7x^{2}+10x-1128=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 7x^{2}+ax+bx-1128. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=7\left(-1128\right)=-7896$$

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 -7896.

$$-1,7896 -2,3948 -3,2632 -4,1974 -6,1316 -7,1128 -8,987 -12,658 -14,564 -21,376 -24,329 -28,282 -42,188 -47,168 -56,141 -84,94$$

Tātaihia te tapeke mō ia takirua.

$$-1+7896=7895 -2+3948=3946 -3+2632=2629 -4+1974=1970 -6+1316=1310 -7+1128=1121 -8+987=979 -12+658=646 -14+564=550 -21+376=355 -24+329=305 -28+282=254 -42+188=146 -47+168=121 -56+141=85 -84+94=10$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=-84 b=94$$

Tuhia anō te 7x^{2}+10x-1128 hei \left(7x^{2}-84x\right)+\left(94x-1128\right).

$$\left(7x^{2}-84x\right)+\left(94x-1128\right)$$

Tauwehea te 7x i te tuatahi me te 94 i te rōpū tuarua.

$$7x\left(x-12\right)+94\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(7x+94\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te 7x+94=0.

$$x=12 x=-\frac{94}{7}$$

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.

$$21x^{2}+30x=3384$$

Me tango 3384 mai i ngā taha e rua o te whārite.

$$21x^{2}+30x-3384=3384-3384$$

Mā te tango i te 3384 i a ia ake anō ka toe ko te 0.

$$21x^{2}+30x-3384=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 21 mō a, 30 mō b, me -3384 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 21\left(-3384\right)}}{2\times 21}$$

Pūrua 30.

$$x=\frac{-30±\sqrt{900-4\times 21\left(-3384\right)}}{2\times 21}$$

Whakareatia -4 ki te 21.

$$x=\frac{-30±\sqrt{900-84\left(-3384\right)}}{2\times 21}$$

Whakareatia -84 ki te -3384.

$$x=\frac{-30±\sqrt{900+284256}}{2\times 21}$$

Tāpiri 900 ki te 284256.

$$x=\frac{-30±\sqrt{285156}}{2\times 21}$$

Tuhia te pūtakerua o te 285156.

$$x=\frac{-30±534}{2\times 21}$$

Whakareatia 2 ki te 21.

$$x=\frac{-30±534}{42}$$

Nā, me whakaoti te whārite x=\frac{-30±534}{42} ina he tāpiri te ±. Tāpiri -30 ki te 534.

$$x=\frac{504}{42}$$

Whakawehe 504 ki te 42.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{-30±534}{42} ina he tango te ±. Tango 534 mai i -30.

$$x=-\frac{564}{42}$$

Whakahekea te hautanga \frac{-564}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{94}{7}$$

Kua oti te whārite te whakatau.

$$x=12 x=-\frac{94}{7}$$

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.

$$21x^{2}+30x=3384$$

Whakawehea ngā taha e rua ki te 21.

$$\frac{21x^{2}+30x}{21}=\frac{3384}{21}$$

Mā te whakawehe ki te 21 ka wetekia te whakareanga ki te 21.

$$x^{2}+\frac{30}{21}x=\frac{3384}{21}$$

Whakahekea te hautanga \frac{30}{21} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{10}{7}x=\frac{3384}{21}$$

Whakahekea te hautanga \frac{3384}{21} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{10}{7}x=\frac{1128}{7}$$

Whakawehea te \frac{10}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{7}. Nā, tāpiria te pūrua o te \frac{5}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{10}{7}x+\left(\frac{5}{7}\right)^{2}=\frac{1128}{7}+\left(\frac{5}{7}\right)^{2}$$

Pūruatia \frac{5}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{10}{7}x+\frac{25}{49}=\frac{1128}{7}+\frac{25}{49}$$

Tāpiri \frac{1128}{7} ki te \frac{25}{49} 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.

$$x^{2}+\frac{10}{7}x+\frac{25}{49}=\frac{7921}{49}$$

Tauwehea x^{2}+\frac{10}{7}x+\frac{25}{49}. 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}.

$$\left(x+\frac{5}{7}\right)^{2}=\frac{7921}{49}$$

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

$$\sqrt{\left(x+\frac{5}{7}\right)^{2}}=\sqrt{\frac{7921}{49}}$$

Whakarūnātia.

$$x+\frac{5}{7}=\frac{89}{7} x+\frac{5}{7}=-\frac{89}{7}$$

Me tango \frac{5}{7} mai i ngā taha e rua o te whārite.

$$x=12 x=-\frac{94}{7}$$