Whakaoti i te (2x+4)^2-5x(7-3x)(7+3x)-(3x-2)^2-40x^2=-205

$$( 2 x + 4 ) ^ { 2 } - 5 x ( 7 - 3 x ) ( 7 + 3 x ) - ( 3 x - 2 ) ^ { 2 } - 40 x ^ { 2 } = - 205$$

Whakaoti mō x

$$x = \frac{\sqrt{1085}}{15} \approx 2.195955879$$

$$x = -\frac{\sqrt{1085}}{15} \approx -2.195955879$$

$$x=1$$

Ngā Upane Otinga

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2x+4\right)^{2}.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-\left(3x-2\right)^{2}-40x^{2}=-205$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3x-2\right)^{2}.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-\left(9x^{2}-12x+4\right)-40x^{2}=-205$$

Hei kimi i te tauaro o 9x^{2}-12x+4, kimihia te tauaro o ia taurangi.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-9x^{2}+12x-4-40x^{2}=-205$$

Pahekotia te -9x^{2} me -40x^{2}, ka -49x^{2}.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-49x^{2}+12x-4=-205$$

Me tāpiri te 205 ki ngā taha e rua.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-49x^{2}+12x-4+205=0$$

Tāpirihia te -4 ki te 205, ka 201.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-49x^{2}+12x+201=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5x ki te 7-3x.

$$4x^{2}+16x+16+\left(-35x+15x^{2}\right)\left(7+3x\right)-49x^{2}+12x+201=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -35x+15x^{2} ki te 7+3x ka whakakotahi i ngā kupu rite.

$$4x^{2}+16x+16-245x+45x^{3}-49x^{2}+12x+201=0$$

Pahekotia te 16x me -245x, ka -229x.

$$4x^{2}-229x+16+45x^{3}-49x^{2}+12x+201=0$$

Pahekotia te 4x^{2} me -49x^{2}, ka -45x^{2}.

$$-45x^{2}-229x+16+45x^{3}+12x+201=0$$

Pahekotia te -229x me 12x, ka -217x.

$$-45x^{2}-217x+16+45x^{3}+201=0$$

Tāpirihia te 16 ki te 201, ka 217.

$$-45x^{2}-217x+217+45x^{3}=0$$

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

$$45x^{3}-45x^{2}-217x+217=0$$

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau 217, ā, ka wehea e q te whakarea arahanga 45. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{217}{45},±\frac{217}{15},±\frac{217}{9},±\frac{217}{5},±\frac{217}{3},±217,±\frac{31}{45},±\frac{31}{15},±\frac{31}{9},±\frac{31}{5},±\frac{31}{3},±31,±\frac{7}{45},±\frac{7}{15},±\frac{7}{9},±\frac{7}{5},±\frac{7}{3},±7,±\frac{1}{45},±\frac{1}{15},±\frac{1}{9},±\frac{1}{5},±\frac{1}{3},±1$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$x=1$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te 45x^{3}-45x^{2}-217x+217 ki te x-1, kia riro ko 45x^{2}-217. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$45x^{2}-217=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 45 mō te a, te 0 mō te b, me te -217 mō te c i te ture pūrua.

$$x=\frac{0±\sqrt{0^{2}-4\times 45\left(-217\right)}}{2\times 45}$$

Mahia ngā tātaitai.

$$x=\frac{0±6\sqrt{1085}}{90}$$

Whakaotia te whārite 45x^{2}-217=0 ina he tōrunga te ±, ina he tōraro te ±.

$$x=-\frac{\sqrt{1085}}{15} x=\frac{\sqrt{1085}}{15}$$

Rārangitia ngā otinga katoa i kitea.

$$x=1 x=-\frac{\sqrt{1085}}{15} x=\frac{\sqrt{1085}}{15}$$