Whakaoti i te 98x^2+40x-30=0

$$98 x ^ { 2 } + 40 x - 30 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{835}-10}{49}\approx 0.385640134$$

$$x=\frac{-\sqrt{835}-10}{49}\approx -0.793803399$$

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.

$$98x^{2}+40x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 98 mō a, 40 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\times 98\left(-30\right)}}{2\times 98}$$

Pūrua 40.

$$x=\frac{-40±\sqrt{1600-4\times 98\left(-30\right)}}{2\times 98}$$

Whakareatia -4 ki te 98.

$$x=\frac{-40±\sqrt{1600-392\left(-30\right)}}{2\times 98}$$

Whakareatia -392 ki te -30.

$$x=\frac{-40±\sqrt{1600+11760}}{2\times 98}$$

Tāpiri 1600 ki te 11760.

$$x=\frac{-40±\sqrt{13360}}{2\times 98}$$

Tuhia te pūtakerua o te 13360.

$$x=\frac{-40±4\sqrt{835}}{2\times 98}$$

Whakareatia 2 ki te 98.

$$x=\frac{-40±4\sqrt{835}}{196}$$

Nā, me whakaoti te whārite x=\frac{-40±4\sqrt{835}}{196} ina he tāpiri te ±. Tāpiri -40 ki te 4\sqrt{835}.

$$x=\frac{4\sqrt{835}-40}{196}$$

Whakawehe -40+4\sqrt{835} ki te 196.

$$x=\frac{\sqrt{835}-10}{49}$$

Nā, me whakaoti te whārite x=\frac{-40±4\sqrt{835}}{196} ina he tango te ±. Tango 4\sqrt{835} mai i -40.

$$x=\frac{-4\sqrt{835}-40}{196}$$

Whakawehe -40-4\sqrt{835} ki te 196.

$$x=\frac{-\sqrt{835}-10}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{835}-10}{49} x=\frac{-\sqrt{835}-10}{49}$$

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.

$$98x^{2}+40x-30=0$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$98x^{2}+40x-30-\left(-30\right)=-\left(-30\right)$$

Mā te tango i te -30 i a ia ake anō ka toe ko te 0.

$$98x^{2}+40x=-\left(-30\right)$$

Tango -30 mai i 0.

$$98x^{2}+40x=30$$

Whakawehea ngā taha e rua ki te 98.

$$\frac{98x^{2}+40x}{98}=\frac{30}{98}$$

Mā te whakawehe ki te 98 ka wetekia te whakareanga ki te 98.

$$x^{2}+\frac{40}{98}x=\frac{30}{98}$$

Whakahekea te hautanga \frac{40}{98} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{20}{49}x=\frac{30}{98}$$

Whakahekea te hautanga \frac{30}{98} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{20}{49}x=\frac{15}{49}$$

Whakawehea te \frac{20}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{10}{49}. Nā, tāpiria te pūrua o te \frac{10}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{20}{49}x+\left(\frac{10}{49}\right)^{2}=\frac{15}{49}+\left(\frac{10}{49}\right)^{2}$$

Pūruatia \frac{10}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{20}{49}x+\frac{100}{2401}=\frac{15}{49}+\frac{100}{2401}$$

Tāpiri \frac{15}{49} ki te \frac{100}{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.

$$x^{2}+\frac{20}{49}x+\frac{100}{2401}=\frac{835}{2401}$$

Tauwehea x^{2}+\frac{20}{49}x+\frac{100}{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}.

$$\left(x+\frac{10}{49}\right)^{2}=\frac{835}{2401}$$

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

$$\sqrt{\left(x+\frac{10}{49}\right)^{2}}=\sqrt{\frac{835}{2401}}$$

Whakarūnātia.

$$x+\frac{10}{49}=\frac{\sqrt{835}}{49} x+\frac{10}{49}=-\frac{\sqrt{835}}{49}$$

Me tango \frac{10}{49} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{835}-10}{49} x=\frac{-\sqrt{835}-10}{49}$$