Whakaoti i te (-2x+9)(-9x+5)+(-9x-5)^2=0

$$( - 2 x + 9 ) ( - 9 x + 5 ) + ( - 9 x - 5 ) ^ { 2 } = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+\sqrt{27719}i}{198}\approx 0.005050505+0.840859798i$$

$$x=\frac{-\sqrt{27719}i+1}{198}\approx 0.005050505-0.840859798i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te -2x+9 ki te -9x+5 ka whakakotahi i ngā kupu rite.

$$18x^{2}-91x+45+\left(-9x-5\right)^{2}=0$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-9x-5\right)^{2}.

$$18x^{2}-91x+45+81x^{2}+90x+25=0$$

Pahekotia te 18x^{2} me 81x^{2}, ka 99x^{2}.

$$99x^{2}-91x+45+90x+25=0$$

Pahekotia te -91x me 90x, ka -x.

$$99x^{2}-x+45+25=0$$

Tāpirihia te 45 ki te 25, ka 70.

$$99x^{2}-x+70=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 99 mō a, -1 mō b, me 70 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 99\times 70}}{2\times 99}$$

Whakareatia -4 ki te 99.

$$x=\frac{-\left(-1\right)±\sqrt{1-396\times 70}}{2\times 99}$$

Whakareatia -396 ki te 70.

$$x=\frac{-\left(-1\right)±\sqrt{1-27720}}{2\times 99}$$

Tāpiri 1 ki te -27720.

$$x=\frac{-\left(-1\right)±\sqrt{-27719}}{2\times 99}$$

Tuhia te pūtakerua o te -27719.

$$x=\frac{-\left(-1\right)±\sqrt{27719}i}{2\times 99}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{27719}i}{2\times 99}$$

Whakareatia 2 ki te 99.

$$x=\frac{1±\sqrt{27719}i}{198}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{27719}i}{198} ina he tāpiri te ±. Tāpiri 1 ki te i\sqrt{27719}.

$$x=\frac{1+\sqrt{27719}i}{198}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{27719}i}{198} ina he tango te ±. Tango i\sqrt{27719} mai i 1.

$$x=\frac{-\sqrt{27719}i+1}{198}$$

Kua oti te whārite te whakatau.

$$x=\frac{1+\sqrt{27719}i}{198} x=\frac{-\sqrt{27719}i+1}{198}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te -2x+9 ki te -9x+5 ka whakakotahi i ngā kupu rite.

$$18x^{2}-91x+45+\left(-9x-5\right)^{2}=0$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-9x-5\right)^{2}.

$$18x^{2}-91x+45+81x^{2}+90x+25=0$$

Pahekotia te 18x^{2} me 81x^{2}, ka 99x^{2}.

$$99x^{2}-91x+45+90x+25=0$$

Pahekotia te -91x me 90x, ka -x.

$$99x^{2}-x+45+25=0$$

Tāpirihia te 45 ki te 25, ka 70.

$$99x^{2}-x+70=0$$

Tangohia te 70 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$99x^{2}-x=-70$$

Whakawehea ngā taha e rua ki te 99.

$$\frac{99x^{2}-x}{99}=-\frac{70}{99}$$

Mā te whakawehe ki te 99 ka wetekia te whakareanga ki te 99.

$$x^{2}-\frac{1}{99}x=-\frac{70}{99}$$

Whakawehea te -\frac{1}{99}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{198}. Nā, tāpiria te pūrua o te -\frac{1}{198} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{99}x+\left(-\frac{1}{198}\right)^{2}=-\frac{70}{99}+\left(-\frac{1}{198}\right)^{2}$$

Pūruatia -\frac{1}{198} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{99}x+\frac{1}{39204}=-\frac{70}{99}+\frac{1}{39204}$$

Tāpiri -\frac{70}{99} ki te \frac{1}{39204} 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.

$$x^{2}-\frac{1}{99}x+\frac{1}{39204}=-\frac{27719}{39204}$$

Tauwehea x^{2}-\frac{1}{99}x+\frac{1}{39204}. 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}.

$$\left(x-\frac{1}{198}\right)^{2}=-\frac{27719}{39204}$$

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

$$\sqrt{\left(x-\frac{1}{198}\right)^{2}}=\sqrt{-\frac{27719}{39204}}$$

Whakarūnātia.

$$x-\frac{1}{198}=\frac{\sqrt{27719}i}{198} x-\frac{1}{198}=-\frac{\sqrt{27719}i}{198}$$

Me tāpiri \frac{1}{198} ki ngā taha e rua o te whārite.

$$x=\frac{1+\sqrt{27719}i}{198} x=\frac{-\sqrt{27719}i+1}{198}$$