Whakaoti i te -left(x+3right)^2-4(3x+1)=0

$$- { \left(x+3 \right) }^{ 2 } -4(3x+1)=0$$

Whakaoti mō x

$$x=2\sqrt{17}-9\approx -0.753788749$$

$$x=-2\sqrt{17}-9\approx -17.246211251$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-\left(x^{2}+6x+9\right)-4\left(3x+1\right)=0$$

Hei kimi i te tauaro o x^{2}+6x+9, kimihia te tauaro o ia taurangi.

$$-x^{2}-6x-9-4\left(3x+1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 3x+1.

$$-x^{2}-6x-9-12x-4=0$$

Pahekotia te -6x me -12x, ka -18x.

$$-x^{2}-18x-9-4=0$$

Tangohia te 4 i te -9, ka -13.

$$-x^{2}-18x-13=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -18 mō b, me -13 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-1\right)\left(-13\right)}}{2\left(-1\right)}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-1\right)\left(-13\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-18\right)±\sqrt{324+4\left(-13\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -13.

$$x=\frac{-\left(-18\right)±\sqrt{324-52}}{2\left(-1\right)}$$

Tāpiri 324 ki te -52.

$$x=\frac{-\left(-18\right)±\sqrt{272}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 272.

$$x=\frac{-\left(-18\right)±4\sqrt{17}}{2\left(-1\right)}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±4\sqrt{17}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{18±4\sqrt{17}}{-2}$$

Nā, me whakaoti te whārite x=\frac{18±4\sqrt{17}}{-2} ina he tāpiri te ±. Tāpiri 18 ki te 4\sqrt{17}.

$$x=\frac{4\sqrt{17}+18}{-2}$$

Whakawehe 18+4\sqrt{17} ki te -2.

$$x=-2\sqrt{17}-9$$

Nā, me whakaoti te whārite x=\frac{18±4\sqrt{17}}{-2} ina he tango te ±. Tango 4\sqrt{17} mai i 18.

$$x=\frac{18-4\sqrt{17}}{-2}$$

Whakawehe 18-4\sqrt{17} ki te -2.

$$x=2\sqrt{17}-9$$

Kua oti te whārite te whakatau.

$$x=-2\sqrt{17}-9 x=2\sqrt{17}-9$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-\left(x^{2}+6x+9\right)-4\left(3x+1\right)=0$$

Hei kimi i te tauaro o x^{2}+6x+9, kimihia te tauaro o ia taurangi.

$$-x^{2}-6x-9-4\left(3x+1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 3x+1.

$$-x^{2}-6x-9-12x-4=0$$

Pahekotia te -6x me -12x, ka -18x.

$$-x^{2}-18x-9-4=0$$

Tangohia te 4 i te -9, ka -13.

$$-x^{2}-18x-13=0$$

Me tāpiri te 13 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-x^{2}-18x=13$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-18x}{-1}=\frac{13}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{18}{-1}\right)x=\frac{13}{-1}$$

Whakawehe -18 ki te -1.

$$x^{2}+18x=\frac{13}{-1}$$

Whakawehe 13 ki te -1.

$$x^{2}+18x=-13$$

Whakawehea te 18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 9. Nā, tāpiria te pūrua o te 9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+18x+9^{2}=-13+9^{2}$$

Pūrua 9.

$$x^{2}+18x+81=-13+81$$

Tāpiri -13 ki te 81.

$$x^{2}+18x+81=68$$

Tauwehea x^{2}+18x+81. 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}.

$$\left(x+9\right)^{2}=68$$

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

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{68}$$

Whakarūnātia.

$$x+9=2\sqrt{17} x+9=-2\sqrt{17}$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x=2\sqrt{17}-9 x=-2\sqrt{17}-9$$