Whakaoti i te (49-x)*(x+1+49)=(49div2)(1+50)

$$(49-x) \times (x+1+49)=(49 \div 2)(1+50)$$

Whakaoti mō x

$$x = \frac{\sqrt{4803} - 1}{2} \approx 34.151839778$$

$$x=\frac{-\sqrt{4803}-1}{2}\approx -35.151839778$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tāpirihia te 1 ki te 49, ka 50.

$$\left(49-x\right)\left(x+50\right)=\frac{49}{2}\left(1+50\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 49-x ki te x+50 ka whakakotahi i ngā kupu rite.

$$-x+2450-x^{2}=\frac{49}{2}\left(1+50\right)$$

Tāpirihia te 1 ki te 50, ka 51.

$$-x+2450-x^{2}=\frac{49}{2}\times 51$$

Whakareatia te \frac{49}{2} ki te 51, ka \frac{2499}{2}.

$$-x+2450-x^{2}=\frac{2499}{2}$$

Tangohia te \frac{2499}{2} mai i ngā taha e rua.

$$-x+2450-x^{2}-\frac{2499}{2}=0$$

Tangohia te \frac{2499}{2} i te 2450, ka \frac{2401}{2}.

$$-x+\frac{2401}{2}-x^{2}=0$$

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.

$$-x^{2}-x+\frac{2401}{2}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -1 mō b, me \frac{2401}{2} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times \frac{2401}{2}}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times \frac{2401}{2}}}{2\left(-1\right)}$$

Whakareatia 4 ki te \frac{2401}{2}.

$$x=\frac{-\left(-1\right)±\sqrt{1+4802}}{2\left(-1\right)}$$

Tāpiri 1 ki te 4802.

$$x=\frac{-\left(-1\right)±\sqrt{4803}}{2\left(-1\right)}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{4803}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{1±\sqrt{4803}}{-2}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{4803}}{-2} ina he tāpiri te ±. Tāpiri 1 ki te \sqrt{4803}.

$$x=\frac{\sqrt{4803}+1}{-2}$$

Whakawehe 1+\sqrt{4803} ki te -2.

$$x=\frac{-\sqrt{4803}-1}{2}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{4803}}{-2} ina he tango te ±. Tango \sqrt{4803} mai i 1.

$$x=\frac{1-\sqrt{4803}}{-2}$$

Whakawehe 1-\sqrt{4803} ki te -2.

$$x=\frac{\sqrt{4803}-1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{4803}-1}{2} x=\frac{\sqrt{4803}-1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tāpirihia te 1 ki te 49, ka 50.

$$\left(49-x\right)\left(x+50\right)=\frac{49}{2}\left(1+50\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 49-x ki te x+50 ka whakakotahi i ngā kupu rite.

$$-x+2450-x^{2}=\frac{49}{2}\left(1+50\right)$$

Tāpirihia te 1 ki te 50, ka 51.

$$-x+2450-x^{2}=\frac{49}{2}\times 51$$

Whakareatia te \frac{49}{2} ki te 51, ka \frac{2499}{2}.

$$-x+2450-x^{2}=\frac{2499}{2}$$

Tangohia te 2450 mai i ngā taha e rua.

$$-x-x^{2}=\frac{2499}{2}-2450$$

Tangohia te 2450 i te \frac{2499}{2}, ka -\frac{2401}{2}.

$$-x-x^{2}=-\frac{2401}{2}$$

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.

$$-x^{2}-x=-\frac{2401}{2}$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-x}{-1}=-\frac{\frac{2401}{2}}{-1}$$

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

$$x^{2}+\left(-\frac{1}{-1}\right)x=-\frac{\frac{2401}{2}}{-1}$$

Whakawehe -1 ki te -1.

$$x^{2}+x=-\frac{\frac{2401}{2}}{-1}$$

Whakawehe -\frac{2401}{2} ki te -1.

$$x^{2}+x=\frac{2401}{2}$$

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.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{2401}{2}+\left(\frac{1}{2}\right)^{2}$$

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

$$x^{2}+x+\frac{1}{4}=\frac{2401}{2}+\frac{1}{4}$$

Tāpiri \frac{2401}{2} ki te \frac{1}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+x+\frac{1}{4}=\frac{4803}{4}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{4803}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{4803}{4}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{\sqrt{4803}}{2} x+\frac{1}{2}=-\frac{\sqrt{4803}}{2}$$

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

$$x=\frac{\sqrt{4803}-1}{2} x=\frac{-\sqrt{4803}-1}{2}$$