Whakaoti i te -4.9x^2+307x+248=0

$$-4.9 { x }^{ 2 } +307x+248=0$$

Whakaoti mō x

$$x = \frac{3 \sqrt{275305} + 1535}{49} \approx 63.450723444$$

$$x=\frac{1535-3\sqrt{275305}}{49}\approx -0.79766222$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$-4.9x^{2}+307x+248=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4.9 mō a, 307 mō b, me 248 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-307±\sqrt{307^{2}-4\left(-4.9\right)\times 248}}{2\left(-4.9\right)}$$

Pūrua 307.

$$x=\frac{-307±\sqrt{94249-4\left(-4.9\right)\times 248}}{2\left(-4.9\right)}$$

Whakareatia -4 ki te -4.9.

$$x=\frac{-307±\sqrt{94249+19.6\times 248}}{2\left(-4.9\right)}$$

Whakareatia 19.6 ki te 248.

$$x=\frac{-307±\sqrt{94249+4860.8}}{2\left(-4.9\right)}$$

Tāpiri 94249 ki te 4860.8.

$$x=\frac{-307±\sqrt{99109.8}}{2\left(-4.9\right)}$$

Tuhia te pūtakerua o te 99109.8.

$$x=\frac{-307±\frac{3\sqrt{275305}}{5}}{2\left(-4.9\right)}$$

Whakareatia 2 ki te -4.9.

$$x=\frac{-307±\frac{3\sqrt{275305}}{5}}{-9.8}$$

Nā, me whakaoti te whārite x=\frac{-307±\frac{3\sqrt{275305}}{5}}{-9.8} ina he tāpiri te ±. Tāpiri -307 ki te \frac{3\sqrt{275305}}{5}.

$$x=\frac{\frac{3\sqrt{275305}}{5}-307}{-9.8}$$

Whakawehe -307+\frac{3\sqrt{275305}}{5} ki te -9.8 mā te whakarea -307+\frac{3\sqrt{275305}}{5} ki te tau huripoki o -9.8.

$$x=\frac{1535-3\sqrt{275305}}{49}$$

Nā, me whakaoti te whārite x=\frac{-307±\frac{3\sqrt{275305}}{5}}{-9.8} ina he tango te ±. Tango \frac{3\sqrt{275305}}{5} mai i -307.

$$x=\frac{-\frac{3\sqrt{275305}}{5}-307}{-9.8}$$

Whakawehe -307-\frac{3\sqrt{275305}}{5} ki te -9.8 mā te whakarea -307-\frac{3\sqrt{275305}}{5} ki te tau huripoki o -9.8.

$$x=\frac{3\sqrt{275305}+1535}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{1535-3\sqrt{275305}}{49} x=\frac{3\sqrt{275305}+1535}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$-4.9x^{2}+307x+248=0$$

Me tango 248 mai i ngā taha e rua o te whārite.

$$-4.9x^{2}+307x+248-248=-248$$

Mā te tango i te 248 i a ia ake anō ka toe ko te 0.

$$-4.9x^{2}+307x=-248$$

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.

$$\frac{-4.9x^{2}+307x}{-4.9}=-\frac{248}{-4.9}$$

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

$$x^{2}+\frac{307}{-4.9}x=-\frac{248}{-4.9}$$

Whakawehe 307 ki te -4.9 mā te whakarea 307 ki te tau huripoki o -4.9.

$$x^{2}-\frac{3070}{49}x=-\frac{248}{-4.9}$$

Whakawehe -248 ki te -4.9 mā te whakarea -248 ki te tau huripoki o -4.9.

$$x^{2}-\frac{3070}{49}x=\frac{2480}{49}$$

Whakawehea te -\frac{3070}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1535}{49}. Nā, tāpiria te pūrua o te -\frac{1535}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3070}{49}x+\left(-\frac{1535}{49}\right)^{2}=\frac{2480}{49}+\left(-\frac{1535}{49}\right)^{2}$$

Pūruatia -\frac{1535}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3070}{49}x+\frac{2356225}{2401}=\frac{2480}{49}+\frac{2356225}{2401}$$

Tāpiri \frac{2480}{49} ki te \frac{2356225}{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.

$$x^{2}-\frac{3070}{49}x+\frac{2356225}{2401}=\frac{2477745}{2401}$$

Tauwehea x^{2}-\frac{3070}{49}x+\frac{2356225}{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}.

$$\left(x-\frac{1535}{49}\right)^{2}=\frac{2477745}{2401}$$

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

$$\sqrt{\left(x-\frac{1535}{49}\right)^{2}}=\sqrt{\frac{2477745}{2401}}$$

Whakarūnātia.

$$x-\frac{1535}{49}=\frac{3\sqrt{275305}}{49} x-\frac{1535}{49}=-\frac{3\sqrt{275305}}{49}$$

Me tāpiri \frac{1535}{49} ki ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{275305}+1535}{49} x=\frac{1535-3\sqrt{275305}}{49}$$