Whakaoti i te -16K^2+164K^2-36K-247/112=0

$$- 16 K ^ { 2 } + 164 K ^ { 2 } - 36 K - \frac { 247 } { 112 } = 0$$

Whakaoti mō K

$$K=\frac{\sqrt{127477}}{2072}+\frac{9}{74}\approx 0.293937844$$

$$K=-\frac{\sqrt{127477}}{2072}+\frac{9}{74}\approx -0.050694601$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te -16K^{2} me 164K^{2}, ka 148K^{2}.

$$148K^{2}-36K-\frac{247}{112}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 148 mō a, -36 mō b, me -\frac{247}{112} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$K=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 148\left(-\frac{247}{112}\right)}}{2\times 148}$$

Pūrua -36.

$$K=\frac{-\left(-36\right)±\sqrt{1296-4\times 148\left(-\frac{247}{112}\right)}}{2\times 148}$$

Whakareatia -4 ki te 148.

$$K=\frac{-\left(-36\right)±\sqrt{1296-592\left(-\frac{247}{112}\right)}}{2\times 148}$$

Whakareatia -592 ki te -\frac{247}{112}.

$$K=\frac{-\left(-36\right)±\sqrt{1296+\frac{9139}{7}}}{2\times 148}$$

Tāpiri 1296 ki te \frac{9139}{7}.

$$K=\frac{-\left(-36\right)±\sqrt{\frac{18211}{7}}}{2\times 148}$$

Tuhia te pūtakerua o te \frac{18211}{7}.

$$K=\frac{-\left(-36\right)±\frac{\sqrt{127477}}{7}}{2\times 148}$$

Ko te tauaro o -36 ko 36.

$$K=\frac{36±\frac{\sqrt{127477}}{7}}{2\times 148}$$

Whakareatia 2 ki te 148.

$$K=\frac{36±\frac{\sqrt{127477}}{7}}{296}$$

Nā, me whakaoti te whārite K=\frac{36±\frac{\sqrt{127477}}{7}}{296} ina he tāpiri te ±. Tāpiri 36 ki te \frac{\sqrt{127477}}{7}.

$$K=\frac{\frac{\sqrt{127477}}{7}+36}{296}$$

Whakawehe 36+\frac{\sqrt{127477}}{7} ki te 296.

$$K=\frac{\sqrt{127477}}{2072}+\frac{9}{74}$$

Nā, me whakaoti te whārite K=\frac{36±\frac{\sqrt{127477}}{7}}{296} ina he tango te ±. Tango \frac{\sqrt{127477}}{7} mai i 36.

$$K=\frac{-\frac{\sqrt{127477}}{7}+36}{296}$$

Whakawehe 36-\frac{\sqrt{127477}}{7} ki te 296.

$$K=-\frac{\sqrt{127477}}{2072}+\frac{9}{74}$$

Kua oti te whārite te whakatau.

$$K=\frac{\sqrt{127477}}{2072}+\frac{9}{74} K=-\frac{\sqrt{127477}}{2072}+\frac{9}{74}$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te -16K^{2} me 164K^{2}, ka 148K^{2}.

$$148K^{2}-36K-\frac{247}{112}=0$$

Me tāpiri te \frac{247}{112} ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$148K^{2}-36K=\frac{247}{112}$$

Whakawehea ngā taha e rua ki te 148.

$$\frac{148K^{2}-36K}{148}=\frac{\frac{247}{112}}{148}$$

Mā te whakawehe ki te 148 ka wetekia te whakareanga ki te 148.

$$K^{2}+\left(-\frac{36}{148}\right)K=\frac{\frac{247}{112}}{148}$$

Whakahekea te hautanga \frac{-36}{148} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$K^{2}-\frac{9}{37}K=\frac{\frac{247}{112}}{148}$$

Whakawehe \frac{247}{112} ki te 148.

$$K^{2}-\frac{9}{37}K=\frac{247}{16576}$$

Whakawehea te -\frac{9}{37}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{74}. Nā, tāpiria te pūrua o te -\frac{9}{74} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$K^{2}-\frac{9}{37}K+\left(-\frac{9}{74}\right)^{2}=\frac{247}{16576}+\left(-\frac{9}{74}\right)^{2}$$

Pūruatia -\frac{9}{74} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$K^{2}-\frac{9}{37}K+\frac{81}{5476}=\frac{247}{16576}+\frac{81}{5476}$$

Tāpiri \frac{247}{16576} ki te \frac{81}{5476} 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.

$$K^{2}-\frac{9}{37}K+\frac{81}{5476}=\frac{18211}{613312}$$

Tauwehea K^{2}-\frac{9}{37}K+\frac{81}{5476}. 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}.

$$\left(K-\frac{9}{74}\right)^{2}=\frac{18211}{613312}$$

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

$$\sqrt{\left(K-\frac{9}{74}\right)^{2}}=\sqrt{\frac{18211}{613312}}$$

Whakarūnātia.

$$K-\frac{9}{74}=\frac{\sqrt{127477}}{2072} K-\frac{9}{74}=-\frac{\sqrt{127477}}{2072}$$

Me tāpiri \frac{9}{74} ki ngā taha e rua o te whārite.

$$K=\frac{\sqrt{127477}}{2072}+\frac{9}{74} K=-\frac{\sqrt{127477}}{2072}+\frac{9}{74}$$