Whakaoti i te 16x^2+10x-9=0

$$16 x ^ { 2 } + 10 x - 9 = 0$$

Whakaoti mō x

$$x = -\frac{9}{8} = -1\frac{1}{8} = -1.125$$

$$x=\frac{1}{2}=0.5$$

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 16x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=16\left(-9\right)=-144$$

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 -144.

$$-1,144 -2,72 -3,48 -4,36 -6,24 -8,18 -9,16 -12,12$$

Tātaihia te tapeke mō ia takirua.

$$-1+144=143 -2+72=70 -3+48=45 -4+36=32 -6+24=18 -8+18=10 -9+16=7 -12+12=0$$

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

$$a=-8 b=18$$

Tuhia anō te 16x^{2}+10x-9 hei \left(16x^{2}-8x\right)+\left(18x-9\right).

$$\left(16x^{2}-8x\right)+\left(18x-9\right)$$

Tauwehea te 8x i te tuatahi me te 9 i te rōpū tuarua.

$$8x\left(2x-1\right)+9\left(2x-1\right)$$

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-1\right)\left(8x+9\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-1=0 me te 8x+9=0.

$$x=\frac{1}{2} x=-\frac{9}{8}$$

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.

$$16x^{2}+10x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, 10 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 16\left(-9\right)}}{2\times 16}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 16\left(-9\right)}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-10±\sqrt{100-64\left(-9\right)}}{2\times 16}$$

Whakareatia -64 ki te -9.

$$x=\frac{-10±\sqrt{100+576}}{2\times 16}$$

Tāpiri 100 ki te 576.

$$x=\frac{-10±\sqrt{676}}{2\times 16}$$

Tuhia te pūtakerua o te 676.

$$x=\frac{-10±26}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{-10±26}{32}$$

Nā, me whakaoti te whārite x=\frac{-10±26}{32} ina he tāpiri te ±. Tāpiri -10 ki te 26.

$$x=\frac{16}{32}$$

Whakahekea te hautanga \frac{16}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-10±26}{32} ina he tango te ±. Tango 26 mai i -10.

$$x=-\frac{36}{32}$$

Whakahekea te hautanga \frac{-36}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{9}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{2} x=-\frac{9}{8}$$

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.

$$16x^{2}+10x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$16x^{2}+10x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$16x^{2}+10x=-\left(-9\right)$$

Tango -9 mai i 0.

$$16x^{2}+10x=9$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}+10x}{16}=\frac{9}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$x^{2}+\frac{10}{16}x=\frac{9}{16}$$

Whakahekea te hautanga \frac{10}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{5}{8}x=\frac{9}{16}$$

Whakawehea te \frac{5}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{16}. Nā, tāpiria te pūrua o te \frac{5}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{8}x+\left(\frac{5}{16}\right)^{2}=\frac{9}{16}+\left(\frac{5}{16}\right)^{2}$$

Pūruatia \frac{5}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{8}x+\frac{25}{256}=\frac{9}{16}+\frac{25}{256}$$

Tāpiri \frac{9}{16} ki te \frac{25}{256} 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.

$$x^{2}+\frac{5}{8}x+\frac{25}{256}=\frac{169}{256}$$

Tauwehea te x^{2}+\frac{5}{8}x+\frac{25}{256}. 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}.

$$\left(x+\frac{5}{16}\right)^{2}=\frac{169}{256}$$

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

$$\sqrt{\left(x+\frac{5}{16}\right)^{2}}=\sqrt{\frac{169}{256}}$$

Whakarūnātia.

$$x+\frac{5}{16}=\frac{13}{16} x+\frac{5}{16}=-\frac{13}{16}$$

Me tango \frac{5}{16} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{2} x=-\frac{9}{8}$$