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

$$- 3 ( 2 x - 1 ) + ( x + 1 ) ( x - 1 ) - 5 ( x + 2 ) = 1$$

Whakaoti mō x

$$x = \frac{\sqrt{157} + 11}{2} \approx 11.764982043$$

$$x=\frac{11-\sqrt{157}}{2}\approx -0.764982043$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 2x-1.

$$-6x+3+\left(x+1\right)\left(x-1\right)-5\left(x+2\right)=1$$

Whakaarohia te \left(x+1\right)\left(x-1\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 1.

$$-6x+3+x^{2}-1-5\left(x+2\right)=1$$

Tangohia te 1 i te 3, ka 2.

$$-6x+2+x^{2}-5\left(x+2\right)=1$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te x+2.

$$-6x+2+x^{2}-5x-10=1$$

Pahekotia te -6x me -5x, ka -11x.

$$-11x+2+x^{2}-10=1$$

Tangohia te 10 i te 2, ka -8.

$$-11x-8+x^{2}=1$$

Tangohia te 1 mai i ngā taha e rua.

$$-11x-8+x^{2}-1=0$$

Tangohia te 1 i te -8, ka -9.

$$-11x-9+x^{2}=0$$

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.

$$x^{2}-11x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -11 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-9\right)}}{2}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-9\right)}}{2}$$

Whakareatia -4 ki te -9.

$$x=\frac{-\left(-11\right)±\sqrt{121+36}}{2}$$

Tāpiri 121 ki te 36.

$$x=\frac{-\left(-11\right)±\sqrt{157}}{2}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±\sqrt{157}}{2}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{157}}{2} ina he tāpiri te ±. Tāpiri 11 ki te \sqrt{157}.

$$x=\frac{\sqrt{157}+11}{2}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{157}}{2} ina he tango te ±. Tango \sqrt{157} mai i 11.

$$x=\frac{11-\sqrt{157}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{157}+11}{2} x=\frac{11-\sqrt{157}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 2x-1.

$$-6x+3+\left(x+1\right)\left(x-1\right)-5\left(x+2\right)=1$$

Whakaarohia te \left(x+1\right)\left(x-1\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 1.

$$-6x+3+x^{2}-1-5\left(x+2\right)=1$$

Tangohia te 1 i te 3, ka 2.

$$-6x+2+x^{2}-5\left(x+2\right)=1$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te x+2.

$$-6x+2+x^{2}-5x-10=1$$

Pahekotia te -6x me -5x, ka -11x.

$$-11x+2+x^{2}-10=1$$

Tangohia te 10 i te 2, ka -8.

$$-11x-8+x^{2}=1$$

Me tāpiri te 8 ki ngā taha e rua.

$$-11x+x^{2}=1+8$$

Tāpirihia te 1 ki te 8, ka 9.

$$-11x+x^{2}=9$$

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.

$$x^{2}-11x=9$$

Whakawehea te -11, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{2}. Nā, tāpiria te pūrua o te -\frac{11}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=9+\left(-\frac{11}{2}\right)^{2}$$

Pūruatia -\frac{11}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-11x+\frac{121}{4}=9+\frac{121}{4}$$

Tāpiri 9 ki te \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{157}{4}$$

Tauwehea x^{2}-11x+\frac{121}{4}. 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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{157}{4}$$

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

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{157}{4}}$$

Whakarūnātia.

$$x-\frac{11}{2}=\frac{\sqrt{157}}{2} x-\frac{11}{2}=-\frac{\sqrt{157}}{2}$$

Me tāpiri \frac{11}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{157}+11}{2} x=\frac{11-\sqrt{157}}{2}$$