Whakaoti i te 28-(x+1)x=3

$$28-(x+1)x=3$$

Whakaoti mō x

$$x = \frac{\sqrt{101} - 1}{2} \approx 4.524937811$$

$$x=\frac{-\sqrt{101}-1}{2}\approx -5.524937811$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$28-\left(x^{2}+x\right)=3$$

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

$$28-x^{2}-x=3$$

Tangohia te 3 mai i ngā taha e rua.

$$28-x^{2}-x-3=0$$

Tangohia te 3 i te 28, ka 25.

$$25-x^{2}-x=0$$

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.

$$-x^{2}-x+25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -1 mō b, me 25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 25}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times 25}}{2\left(-1\right)}$$

Whakareatia 4 ki te 25.

$$x=\frac{-\left(-1\right)±\sqrt{1+100}}{2\left(-1\right)}$$

Tāpiri 1 ki te 100.

$$x=\frac{-\left(-1\right)±\sqrt{101}}{2\left(-1\right)}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{101}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{1±\sqrt{101}}{-2}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{101}}{-2} ina he tāpiri te ±. Tāpiri 1 ki te \sqrt{101}.

$$x=\frac{\sqrt{101}+1}{-2}$$

Whakawehe 1+\sqrt{101} ki te -2.

$$x=\frac{-\sqrt{101}-1}{2}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{101}}{-2} ina he tango te ±. Tango \sqrt{101} mai i 1.

$$x=\frac{1-\sqrt{101}}{-2}$$

Whakawehe 1-\sqrt{101} ki te -2.

$$x=\frac{\sqrt{101}-1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{101}-1}{2} x=\frac{\sqrt{101}-1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$28-\left(x^{2}+x\right)=3$$

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

$$28-x^{2}-x=3$$

Tangohia te 28 mai i ngā taha e rua.

$$-x^{2}-x=3-28$$

Tangohia te 28 i te 3, ka -25.

$$-x^{2}-x=-25$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-x}{-1}=-\frac{25}{-1}$$

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

$$x^{2}+\left(-\frac{1}{-1}\right)x=-\frac{25}{-1}$$

Whakawehe -1 ki te -1.

$$x^{2}+x=-\frac{25}{-1}$$

Whakawehe -25 ki te -1.

$$x^{2}+x=25$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=25+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=25+\frac{1}{4}$$

Tāpiri 25 ki te \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{101}{4}$$

Tauwehea x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{101}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{101}{4}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{\sqrt{101}}{2} x+\frac{1}{2}=-\frac{\sqrt{101}}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{101}-1}{2} x=\frac{-\sqrt{101}-1}{2}$$