Whakaoti i te 11=1+20x-4.9x^2

$$11=1+20x-4.9 { x }^{ 2 }$$

Whakaoti mō x

$$x = \frac{10 \sqrt{51} + 100}{49} \approx 3.4982507$$

$$x=\frac{100-10\sqrt{51}}{49}\approx 0.583381953$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1+20x-4.9x^{2}=11$$

Tangohia te 11 mai i ngā taha e rua.

$$1+20x-4.9x^{2}-11=0$$

Tangohia te 11 i te 1, ka -10.

$$-10+20x-4.9x^{2}=0$$

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.

$$-4.9x^{2}+20x-10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4.9 mō a, 20 mō b, me -10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-4.9\right)\left(-10\right)}}{2\left(-4.9\right)}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\left(-4.9\right)\left(-10\right)}}{2\left(-4.9\right)}$$

Whakareatia -4 ki te -4.9.

$$x=\frac{-20±\sqrt{400+19.6\left(-10\right)}}{2\left(-4.9\right)}$$

Whakareatia 19.6 ki te -10.

$$x=\frac{-20±\sqrt{400-196}}{2\left(-4.9\right)}$$

Tāpiri 400 ki te -196.

$$x=\frac{-20±\sqrt{204}}{2\left(-4.9\right)}$$

Tuhia te pūtakerua o te 204.

$$x=\frac{-20±2\sqrt{51}}{2\left(-4.9\right)}$$

Whakareatia 2 ki te -4.9.

$$x=\frac{-20±2\sqrt{51}}{-9.8}$$

Nā, me whakaoti te whārite x=\frac{-20±2\sqrt{51}}{-9.8} ina he tāpiri te ±. Tāpiri -20 ki te 2\sqrt{51}.

$$x=\frac{2\sqrt{51}-20}{-9.8}$$

Whakawehe -20+2\sqrt{51} ki te -9.8 mā te whakarea -20+2\sqrt{51} ki te tau huripoki o -9.8.

$$x=\frac{100-10\sqrt{51}}{49}$$

Nā, me whakaoti te whārite x=\frac{-20±2\sqrt{51}}{-9.8} ina he tango te ±. Tango 2\sqrt{51} mai i -20.

$$x=\frac{-2\sqrt{51}-20}{-9.8}$$

Whakawehe -20-2\sqrt{51} ki te -9.8 mā te whakarea -20-2\sqrt{51} ki te tau huripoki o -9.8.

$$x=\frac{10\sqrt{51}+100}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{100-10\sqrt{51}}{49} x=\frac{10\sqrt{51}+100}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1+20x-4.9x^{2}=11$$

Tangohia te 1 mai i ngā taha e rua.

$$20x-4.9x^{2}=11-1$$

Tangohia te 1 i te 11, ka 10.

$$20x-4.9x^{2}=10$$

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.

$$-4.9x^{2}+20x=10$$

Whakawehea ngā taha e rua o te whārite ki te -4.9, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-4.9x^{2}+20x}{-4.9}=\frac{10}{-4.9}$$

Mā te whakawehe ki te -4.9 ka wetekia te whakareanga ki te -4.9.

$$x^{2}+\frac{20}{-4.9}x=\frac{10}{-4.9}$$

Whakawehe 20 ki te -4.9 mā te whakarea 20 ki te tau huripoki o -4.9.

$$x^{2}-\frac{200}{49}x=\frac{10}{-4.9}$$

Whakawehe 10 ki te -4.9 mā te whakarea 10 ki te tau huripoki o -4.9.

$$x^{2}-\frac{200}{49}x=-\frac{100}{49}$$

Whakawehea te -\frac{200}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{100}{49}. Nā, tāpiria te pūrua o te -\frac{100}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{200}{49}x+\left(-\frac{100}{49}\right)^{2}=-\frac{100}{49}+\left(-\frac{100}{49}\right)^{2}$$

Pūruatia -\frac{100}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{200}{49}x+\frac{10000}{2401}=-\frac{100}{49}+\frac{10000}{2401}$$

Tāpiri -\frac{100}{49} ki te \frac{10000}{2401} 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.

$$x^{2}-\frac{200}{49}x+\frac{10000}{2401}=\frac{5100}{2401}$$

Tauwehea x^{2}-\frac{200}{49}x+\frac{10000}{2401}. 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}.

$$\left(x-\frac{100}{49}\right)^{2}=\frac{5100}{2401}$$

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

$$\sqrt{\left(x-\frac{100}{49}\right)^{2}}=\sqrt{\frac{5100}{2401}}$$

Whakarūnātia.

$$x-\frac{100}{49}=\frac{10\sqrt{51}}{49} x-\frac{100}{49}=-\frac{10\sqrt{51}}{49}$$

Me tāpiri \frac{100}{49} ki ngā taha e rua o te whārite.

$$x=\frac{10\sqrt{51}+100}{49} x=\frac{100-10\sqrt{51}}{49}$$