Whakaoti i te 4-13x-7x^2=0

$$4 - 13 x - 7 x ^ { 2 } = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{281}-13}{14}\approx 0.268789615$$

$$x=\frac{-\sqrt{281}-13}{14}\approx -2.125932472$$

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.

$$-7x^{2}-13x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -7 mō a, -13 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-7\right)\times 4}}{2\left(-7\right)}$$

Pūrua -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-7\right)\times 4}}{2\left(-7\right)}$$

Whakareatia -4 ki te -7.

$$x=\frac{-\left(-13\right)±\sqrt{169+28\times 4}}{2\left(-7\right)}$$

Whakareatia 28 ki te 4.

$$x=\frac{-\left(-13\right)±\sqrt{169+112}}{2\left(-7\right)}$$

Tāpiri 169 ki te 112.

$$x=\frac{-\left(-13\right)±\sqrt{281}}{2\left(-7\right)}$$

Ko te tauaro o -13 ko 13.

$$x=\frac{13±\sqrt{281}}{2\left(-7\right)}$$

Whakareatia 2 ki te -7.

$$x=\frac{13±\sqrt{281}}{-14}$$

Nā, me whakaoti te whārite x=\frac{13±\sqrt{281}}{-14} ina he tāpiri te ±. Tāpiri 13 ki te \sqrt{281}.

$$x=\frac{\sqrt{281}+13}{-14}$$

Whakawehe 13+\sqrt{281} ki te -14.

$$x=\frac{-\sqrt{281}-13}{14}$$

Nā, me whakaoti te whārite x=\frac{13±\sqrt{281}}{-14} ina he tango te ±. Tango \sqrt{281} mai i 13.

$$x=\frac{13-\sqrt{281}}{-14}$$

Whakawehe 13-\sqrt{281} ki te -14.

$$x=\frac{\sqrt{281}-13}{14}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{281}-13}{14} x=\frac{\sqrt{281}-13}{14}$$

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.

$$-7x^{2}-13x+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-7x^{2}-13x+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$-7x^{2}-13x=-4$$

Whakawehea ngā taha e rua ki te -7.

$$\frac{-7x^{2}-13x}{-7}=-\frac{4}{-7}$$

Mā te whakawehe ki te -7 ka wetekia te whakareanga ki te -7.

$$x^{2}+\left(-\frac{13}{-7}\right)x=-\frac{4}{-7}$$

Whakawehe -13 ki te -7.

$$x^{2}+\frac{13}{7}x=-\frac{4}{-7}$$

Whakawehe -4 ki te -7.

$$x^{2}+\frac{13}{7}x=\frac{4}{7}$$

Whakawehea te \frac{13}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{13}{14}. Nā, tāpiria te pūrua o te \frac{13}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{13}{7}x+\left(\frac{13}{14}\right)^{2}=\frac{4}{7}+\left(\frac{13}{14}\right)^{2}$$

Pūruatia \frac{13}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{13}{7}x+\frac{169}{196}=\frac{4}{7}+\frac{169}{196}$$

Tāpiri \frac{4}{7} ki te \frac{169}{196} 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.

$$x^{2}+\frac{13}{7}x+\frac{169}{196}=\frac{281}{196}$$

Tauwehea x^{2}+\frac{13}{7}x+\frac{169}{196}. 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}.

$$\left(x+\frac{13}{14}\right)^{2}=\frac{281}{196}$$

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

$$\sqrt{\left(x+\frac{13}{14}\right)^{2}}=\sqrt{\frac{281}{196}}$$

Whakarūnātia.

$$x+\frac{13}{14}=\frac{\sqrt{281}}{14} x+\frac{13}{14}=-\frac{\sqrt{281}}{14}$$

Me tango \frac{13}{14} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{281}-13}{14} x=\frac{-\sqrt{281}-13}{14}$$