Whakaoti i te (2x+3)(x-2)-x(x+1)=0

$$( 2 x + 3 ) ( x - 2 ) - x ( x + 1 ) = 0$$

Whakaoti mō x

$$x=\sqrt{7}+1\approx 3.645751311$$

$$x=1-\sqrt{7}\approx -1.645751311$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+3 ki te x-2 ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+1.

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

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

$$2x^{2}-x-6-x^{2}-x=0$$

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

$$x^{2}-x-6-x=0$$

Pahekotia te -x me -x, ka -2x.

$$x^{2}-2x-6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -2 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-6\right)}}{2}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-6\right)}}{2}$$

Whakareatia -4 ki te -6.

$$x=\frac{-\left(-2\right)±\sqrt{4+24}}{2}$$

Tāpiri 4 ki te 24.

$$x=\frac{-\left(-2\right)±\sqrt{28}}{2}$$

Tuhia te pūtakerua o te 28.

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

Ko te tauaro o -2 ko 2.

$$x=\frac{2±2\sqrt{7}}{2}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{7}}{2} ina he tāpiri te ±. Tāpiri 2 ki te 2\sqrt{7}.

$$x=\frac{2\sqrt{7}+2}{2}$$

Whakawehe 2+2\sqrt{7} ki te 2.

$$x=\sqrt{7}+1$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{7}}{2} ina he tango te ±. Tango 2\sqrt{7} mai i 2.

$$x=\frac{2-2\sqrt{7}}{2}$$

Whakawehe 2-2\sqrt{7} ki te 2.

$$x=1-\sqrt{7}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{7}+1 x=1-\sqrt{7}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+3 ki te x-2 ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+1.

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

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

$$2x^{2}-x-6-x^{2}-x=0$$

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

$$x^{2}-x-6-x=0$$

Pahekotia te -x me -x, ka -2x.

$$x^{2}-2x-6=0$$

Me tāpiri te 6 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}-2x=6$$

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

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

Tāpiri 6 ki te 1.

$$x^{2}-2x+1=7$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=7$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{7}$$

Whakarūnātia.

$$x-1=\sqrt{7} x-1=-\sqrt{7}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=\sqrt{7}+1 x=1-\sqrt{7}$$