Whakaoti i te x^3-7x^2+5=x(x^2-1)+3x^2-2

$$x ^ { 3 } - 7 x ^ { 2 } + 5 = x ( x ^ { 2 } - 1 ) + 3 x ^ { 2 } - 2$$

Whakaoti mō x

$$x=\frac{\sqrt{281}+1}{20}\approx 0.888152731$$

$$x=\frac{1-\sqrt{281}}{20}\approx -0.788152731$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x^{2}-1.

$$x^{3}-7x^{2}+5=x^{3}-x+3x^{2}-2$$

Tangohia te x^{3} mai i ngā taha e rua.

$$x^{3}-7x^{2}+5-x^{3}=-x+3x^{2}-2$$

Pahekotia te x^{3} me -x^{3}, ka 0.

$$-7x^{2}+5=-x+3x^{2}-2$$

Me tāpiri te x ki ngā taha e rua.

$$-7x^{2}+5+x=3x^{2}-2$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$-7x^{2}+5+x-3x^{2}=-2$$

Pahekotia te -7x^{2} me -3x^{2}, ka -10x^{2}.

$$-10x^{2}+5+x=-2$$

Me tāpiri te 2 ki ngā taha e rua.

$$-10x^{2}+5+x+2=0$$

Tāpirihia te 5 ki te 2, ka 7.

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

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -10 mō a, 1 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-10\right)\times 7}}{2\left(-10\right)}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-10\right)\times 7}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$x=\frac{-1±\sqrt{1+40\times 7}}{2\left(-10\right)}$$

Whakareatia 40 ki te 7.

$$x=\frac{-1±\sqrt{1+280}}{2\left(-10\right)}$$

Tāpiri 1 ki te 280.

$$x=\frac{-1±\sqrt{281}}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$x=\frac{-1±\sqrt{281}}{-20}$$

Nā, me whakaoti te whārite x=\frac{-1±\sqrt{281}}{-20} ina he tāpiri te ±. Tāpiri -1 ki te \sqrt{281}.

$$x=\frac{\sqrt{281}-1}{-20}$$

Whakawehe -1+\sqrt{281} ki te -20.

$$x=\frac{1-\sqrt{281}}{20}$$

Nā, me whakaoti te whārite x=\frac{-1±\sqrt{281}}{-20} ina he tango te ±. Tango \sqrt{281} mai i -1.

$$x=\frac{-\sqrt{281}-1}{-20}$$

Whakawehe -1-\sqrt{281} ki te -20.

$$x=\frac{\sqrt{281}+1}{20}$$

Kua oti te whārite te whakatau.

$$x=\frac{1-\sqrt{281}}{20} x=\frac{\sqrt{281}+1}{20}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x^{2}-1.

$$x^{3}-7x^{2}+5=x^{3}-x+3x^{2}-2$$

Tangohia te x^{3} mai i ngā taha e rua.

$$x^{3}-7x^{2}+5-x^{3}=-x+3x^{2}-2$$

Pahekotia te x^{3} me -x^{3}, ka 0.

$$-7x^{2}+5=-x+3x^{2}-2$$

Me tāpiri te x ki ngā taha e rua.

$$-7x^{2}+5+x=3x^{2}-2$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$-7x^{2}+5+x-3x^{2}=-2$$

Pahekotia te -7x^{2} me -3x^{2}, ka -10x^{2}.

$$-10x^{2}+5+x=-2$$

Tangohia te 5 mai i ngā taha e rua.

$$-10x^{2}+x=-2-5$$

Tangohia te 5 i te -2, ka -7.

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

Whakawehea ngā taha e rua ki te -10.

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

Mā te whakawehe ki te -10 ka wetekia te whakareanga ki te -10.

$$x^{2}+\frac{1}{-10}x=-\frac{7}{-10}$$

Whakawehe 1 ki te -10.

$$x^{2}-\frac{1}{10}x=-\frac{7}{-10}$$

Whakawehe -7 ki te -10.

$$x^{2}-\frac{1}{10}x=\frac{7}{10}$$

Whakawehea te -\frac{1}{10}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{20}. Nā, tāpiria te pūrua o te -\frac{1}{20} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{10}x+\left(-\frac{1}{20}\right)^{2}=\frac{7}{10}+\left(-\frac{1}{20}\right)^{2}$$

Pūruatia -\frac{1}{20} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{10}x+\frac{1}{400}=\frac{7}{10}+\frac{1}{400}$$

Tāpiri \frac{7}{10} ki te \frac{1}{400} 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.

$$x^{2}-\frac{1}{10}x+\frac{1}{400}=\frac{281}{400}$$

Tauwehea x^{2}-\frac{1}{10}x+\frac{1}{400}. 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}.

$$\left(x-\frac{1}{20}\right)^{2}=\frac{281}{400}$$

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

$$\sqrt{\left(x-\frac{1}{20}\right)^{2}}=\sqrt{\frac{281}{400}}$$

Whakarūnātia.

$$x-\frac{1}{20}=\frac{\sqrt{281}}{20} x-\frac{1}{20}=-\frac{\sqrt{281}}{20}$$

Me tāpiri \frac{1}{20} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{281}+1}{20} x=\frac{1-\sqrt{281}}{20}$$