Whakaoti i te 4x^2+13x+5=0

$$4 { x }^{ 2 } +13x+5=0$$

Whakaoti mō x

$$x=\frac{\sqrt{89}-13}{8}\approx -0.445752358$$

$$x=\frac{-\sqrt{89}-13}{8}\approx -2.804247642$$

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.

$$4x^{2}+13x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 13 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 4\times 5}}{2\times 4}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\times 4\times 5}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-13±\sqrt{169-16\times 5}}{2\times 4}$$

Whakareatia -16 ki te 5.

$$x=\frac{-13±\sqrt{169-80}}{2\times 4}$$

Tāpiri 169 ki te -80.

$$x=\frac{-13±\sqrt{89}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-13±\sqrt{89}}{8}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{89}}{8} ina he tāpiri te ±. Tāpiri -13 ki te \sqrt{89}.

$$x=\frac{\sqrt{89}-13}{8}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{89}}{8} ina he tango te ±. Tango \sqrt{89} mai i -13.

$$x=\frac{-\sqrt{89}-13}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{89}-13}{8} x=\frac{-\sqrt{89}-13}{8}$$

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.

$$4x^{2}+13x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$4x^{2}+13x+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$4x^{2}+13x=-5$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+13x}{4}=-\frac{5}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\frac{13}{4}x=-\frac{5}{4}$$

Whakawehea te \frac{13}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{13}{8}. Nā, tāpiria te pūrua o te \frac{13}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{13}{4}x+\left(\frac{13}{8}\right)^{2}=-\frac{5}{4}+\left(\frac{13}{8}\right)^{2}$$

Pūruatia \frac{13}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=-\frac{5}{4}+\frac{169}{64}$$

Tāpiri -\frac{5}{4} ki te \frac{169}{64} 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.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=\frac{89}{64}$$

Tauwehea x^{2}+\frac{13}{4}x+\frac{169}{64}. 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}.

$$\left(x+\frac{13}{8}\right)^{2}=\frac{89}{64}$$

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

$$\sqrt{\left(x+\frac{13}{8}\right)^{2}}=\sqrt{\frac{89}{64}}$$

Whakarūnātia.

$$x+\frac{13}{8}=\frac{\sqrt{89}}{8} x+\frac{13}{8}=-\frac{\sqrt{89}}{8}$$

Me tango \frac{13}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{89}-13}{8} x=\frac{-\sqrt{89}-13}{8}$$