Whakaoti i te 23.15=24.6x-0.5*9.8x^2

$$23.15=24.6x-0.5 \times 9.8 { x }^{ 2 }$$

Whakaoti mō x

$$x=\frac{\sqrt{15142}}{98}+\frac{123}{49}\approx 3.765845247$$

$$x=-\frac{\sqrt{15142}}{98}+\frac{123}{49}\approx 1.254562917$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 0.5 ki te 9.8, ka 4.9.

$$23.15=24.6x-4.9x^{2}$$

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

$$24.6x-4.9x^{2}=23.15$$

Tangohia te 23.15 mai i ngā taha e rua.

$$24.6x-4.9x^{2}-23.15=0$$

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.

$$-4.9x^{2}+24.6x-23.15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4.9 mō a, 24.6 mō b, me -23.15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24.6±\sqrt{24.6^{2}-4\left(-4.9\right)\left(-23.15\right)}}{2\left(-4.9\right)}$$

Pūruatia 24.6 mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-24.6±\sqrt{605.16-4\left(-4.9\right)\left(-23.15\right)}}{2\left(-4.9\right)}$$

Whakareatia -4 ki te -4.9.

$$x=\frac{-24.6±\sqrt{605.16+19.6\left(-23.15\right)}}{2\left(-4.9\right)}$$

Whakareatia 19.6 ki te -23.15 mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-24.6±\sqrt{605.16-453.74}}{2\left(-4.9\right)}$$

Tāpiri 605.16 ki te -453.74 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.

$$x=\frac{-24.6±\sqrt{151.42}}{2\left(-4.9\right)}$$

Tuhia te pūtakerua o te 151.42.

$$x=\frac{-24.6±\frac{\sqrt{15142}}{10}}{2\left(-4.9\right)}$$

Whakareatia 2 ki te -4.9.

$$x=\frac{-24.6±\frac{\sqrt{15142}}{10}}{-9.8}$$

Nā, me whakaoti te whārite x=\frac{-24.6±\frac{\sqrt{15142}}{10}}{-9.8} ina he tāpiri te ±. Tāpiri -24.6 ki te \frac{\sqrt{15142}}{10}.

$$x=\frac{\frac{\sqrt{15142}}{10}-\frac{123}{5}}{-9.8}$$

Whakawehe -\frac{123}{5}+\frac{\sqrt{15142}}{10} ki te -9.8 mā te whakarea -\frac{123}{5}+\frac{\sqrt{15142}}{10} ki te tau huripoki o -9.8.

$$x=-\frac{\sqrt{15142}}{98}+\frac{123}{49}$$

Nā, me whakaoti te whārite x=\frac{-24.6±\frac{\sqrt{15142}}{10}}{-9.8} ina he tango te ±. Tango \frac{\sqrt{15142}}{10} mai i -24.6.

$$x=\frac{-\frac{\sqrt{15142}}{10}-\frac{123}{5}}{-9.8}$$

Whakawehe -\frac{123}{5}-\frac{\sqrt{15142}}{10} ki te -9.8 mā te whakarea -\frac{123}{5}-\frac{\sqrt{15142}}{10} ki te tau huripoki o -9.8.

$$x=\frac{\sqrt{15142}}{98}+\frac{123}{49}$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{15142}}{98}+\frac{123}{49} x=\frac{\sqrt{15142}}{98}+\frac{123}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 0.5 ki te 9.8, ka 4.9.

$$23.15=24.6x-4.9x^{2}$$

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

$$24.6x-4.9x^{2}=23.15$$

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.

$$-4.9x^{2}+24.6x=23.15$$

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.

$$\frac{-4.9x^{2}+24.6x}{-4.9}=\frac{23.15}{-4.9}$$

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

$$x^{2}+\frac{24.6}{-4.9}x=\frac{23.15}{-4.9}$$

Whakawehe 24.6 ki te -4.9 mā te whakarea 24.6 ki te tau huripoki o -4.9.

$$x^{2}-\frac{246}{49}x=\frac{23.15}{-4.9}$$

Whakawehe 23.15 ki te -4.9 mā te whakarea 23.15 ki te tau huripoki o -4.9.

$$x^{2}-\frac{246}{49}x=-\frac{463}{98}$$

Whakawehea te -\frac{246}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{123}{49}. Nā, tāpiria te pūrua o te -\frac{123}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{246}{49}x+\left(-\frac{123}{49}\right)^{2}=-\frac{463}{98}+\left(-\frac{123}{49}\right)^{2}$$

Pūruatia -\frac{123}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{246}{49}x+\frac{15129}{2401}=-\frac{463}{98}+\frac{15129}{2401}$$

Tāpiri -\frac{463}{98} ki te \frac{15129}{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.

$$x^{2}-\frac{246}{49}x+\frac{15129}{2401}=\frac{7571}{4802}$$

Tauwehea x^{2}-\frac{246}{49}x+\frac{15129}{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}.

$$\left(x-\frac{123}{49}\right)^{2}=\frac{7571}{4802}$$

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

$$\sqrt{\left(x-\frac{123}{49}\right)^{2}}=\sqrt{\frac{7571}{4802}}$$

Whakarūnātia.

$$x-\frac{123}{49}=\frac{\sqrt{15142}}{98} x-\frac{123}{49}=-\frac{\sqrt{15142}}{98}$$

Me tāpiri \frac{123}{49} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{15142}}{98}+\frac{123}{49} x=-\frac{\sqrt{15142}}{98}+\frac{123}{49}$$