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

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

Whakaoti mō x

$$x = \frac{\sqrt{21} + 1}{2} \approx 2.791287847$$

$$x=\frac{1-\sqrt{21}}{2}\approx -1.791287847$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te 2-x ka whakakotahi i ngā kupu rite.

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

Tāpirihia te 2 ki te 3, ka 5.

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

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 1 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-1±\sqrt{1+4\times 5}}{2\left(-1\right)}$$

Whakareatia 4 ki te 5.

$$x=\frac{-1±\sqrt{1+20}}{2\left(-1\right)}$$

Tāpiri 1 ki te 20.

$$x=\frac{-1±\sqrt{21}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-1±\sqrt{21}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-1±\sqrt{21}}{-2} ina he tāpiri te ±. Tāpiri -1 ki te \sqrt{21}.

$$x=\frac{\sqrt{21}-1}{-2}$$

Whakawehe -1+\sqrt{21} ki te -2.

$$x=\frac{1-\sqrt{21}}{2}$$

Nā, me whakaoti te whārite x=\frac{-1±\sqrt{21}}{-2} ina he tango te ±. Tango \sqrt{21} mai i -1.

$$x=\frac{-\sqrt{21}-1}{-2}$$

Whakawehe -1-\sqrt{21} ki te -2.

$$x=\frac{\sqrt{21}+1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{1-\sqrt{21}}{2} x=\frac{\sqrt{21}+1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te 2-x ka whakakotahi i ngā kupu rite.

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

Tāpirihia te 2 ki te 3, ka 5.

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

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

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.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+x}{-1}=-\frac{5}{-1}$$

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

$$x^{2}+\frac{1}{-1}x=-\frac{5}{-1}$$

Whakawehe 1 ki te -1.

$$x^{2}-x=-\frac{5}{-1}$$

Whakawehe -5 ki te -1.

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

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.

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

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

$$x^{2}-x+\frac{1}{4}=5+\frac{1}{4}$$

Tāpiri 5 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{21}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{21}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{\sqrt{21}}{2} x-\frac{1}{2}=-\frac{\sqrt{21}}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{21}+1}{2} x=\frac{1-\sqrt{21}}{2}$$