Whakaoti i te 49x^2+2x-15=0

$$49 { x }^{ 2 } +2x-15=0$$

Whakaoti mō x

$$x=\frac{4\sqrt{46}-1}{49}\approx 0.533251427$$

$$x=\frac{-4\sqrt{46}-1}{49}\approx -0.574067754$$

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.

$$49x^{2}+2x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 49 mō a, 2 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 49\left(-15\right)}}{2\times 49}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\times 49\left(-15\right)}}{2\times 49}$$

Whakareatia -4 ki te 49.

$$x=\frac{-2±\sqrt{4-196\left(-15\right)}}{2\times 49}$$

Whakareatia -196 ki te -15.

$$x=\frac{-2±\sqrt{4+2940}}{2\times 49}$$

Tāpiri 4 ki te 2940.

$$x=\frac{-2±\sqrt{2944}}{2\times 49}$$

Tuhia te pūtakerua o te 2944.

$$x=\frac{-2±8\sqrt{46}}{2\times 49}$$

Whakareatia 2 ki te 49.

$$x=\frac{-2±8\sqrt{46}}{98}$$

Nā, me whakaoti te whārite x=\frac{-2±8\sqrt{46}}{98} ina he tāpiri te ±. Tāpiri -2 ki te 8\sqrt{46}.

$$x=\frac{8\sqrt{46}-2}{98}$$

Whakawehe -2+8\sqrt{46} ki te 98.

$$x=\frac{4\sqrt{46}-1}{49}$$

Nā, me whakaoti te whārite x=\frac{-2±8\sqrt{46}}{98} ina he tango te ±. Tango 8\sqrt{46} mai i -2.

$$x=\frac{-8\sqrt{46}-2}{98}$$

Whakawehe -2-8\sqrt{46} ki te 98.

$$x=\frac{-4\sqrt{46}-1}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{4\sqrt{46}-1}{49} x=\frac{-4\sqrt{46}-1}{49}$$

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.

$$49x^{2}+2x-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$49x^{2}+2x-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$49x^{2}+2x=-\left(-15\right)$$

Tango -15 mai i 0.

$$49x^{2}+2x=15$$

Whakawehea ngā taha e rua ki te 49.

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

Mā te whakawehe ki te 49 ka wetekia te whakareanga ki te 49.

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

Whakawehea te \frac{2}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{49}. Nā, tāpiria te pūrua o te \frac{1}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{2}{49}x+\left(\frac{1}{49}\right)^{2}=\frac{15}{49}+\left(\frac{1}{49}\right)^{2}$$

Pūruatia \frac{1}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{2}{49}x+\frac{1}{2401}=\frac{15}{49}+\frac{1}{2401}$$

Tāpiri \frac{15}{49} ki te \frac{1}{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.

$$x^{2}+\frac{2}{49}x+\frac{1}{2401}=\frac{736}{2401}$$

Tauwehea x^{2}+\frac{2}{49}x+\frac{1}{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}.

$$\left(x+\frac{1}{49}\right)^{2}=\frac{736}{2401}$$

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

$$\sqrt{\left(x+\frac{1}{49}\right)^{2}}=\sqrt{\frac{736}{2401}}$$

Whakarūnātia.

$$x+\frac{1}{49}=\frac{4\sqrt{46}}{49} x+\frac{1}{49}=-\frac{4\sqrt{46}}{49}$$

Me tango \frac{1}{49} mai i ngā taha e rua o te whārite.

$$x=\frac{4\sqrt{46}-1}{49} x=\frac{-4\sqrt{46}-1}{49}$$