Whakaoti i te 3x^2-52x+48=0

$$3 { x }^{ 2 } -52x+48 = 0$$

Whakaoti mō x

$$x = \frac{2 \sqrt{133} + 26}{3} \approx 16.35504173$$

$$x=\frac{26-2\sqrt{133}}{3}\approx 0.978291604$$

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.

$$3x^{2}-52x+48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -52 mō b, me 48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 3\times 48}}{2\times 3}$$

Pūrua -52.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 3\times 48}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-52\right)±\sqrt{2704-12\times 48}}{2\times 3}$$

Whakareatia -12 ki te 48.

$$x=\frac{-\left(-52\right)±\sqrt{2704-576}}{2\times 3}$$

Tāpiri 2704 ki te -576.

$$x=\frac{-\left(-52\right)±\sqrt{2128}}{2\times 3}$$

Tuhia te pūtakerua o te 2128.

$$x=\frac{-\left(-52\right)±4\sqrt{133}}{2\times 3}$$

Ko te tauaro o -52 ko 52.

$$x=\frac{52±4\sqrt{133}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{52±4\sqrt{133}}{6}$$

Nā, me whakaoti te whārite x=\frac{52±4\sqrt{133}}{6} ina he tāpiri te ±. Tāpiri 52 ki te 4\sqrt{133}.

$$x=\frac{4\sqrt{133}+52}{6}$$

Whakawehe 52+4\sqrt{133} ki te 6.

$$x=\frac{2\sqrt{133}+26}{3}$$

Nā, me whakaoti te whārite x=\frac{52±4\sqrt{133}}{6} ina he tango te ±. Tango 4\sqrt{133} mai i 52.

$$x=\frac{52-4\sqrt{133}}{6}$$

Whakawehe 52-4\sqrt{133} ki te 6.

$$x=\frac{26-2\sqrt{133}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{133}+26}{3} x=\frac{26-2\sqrt{133}}{3}$$

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.

$$3x^{2}-52x+48=0$$

Me tango 48 mai i ngā taha e rua o te whārite.

$$3x^{2}-52x+48-48=-48$$

Mā te tango i te 48 i a ia ake anō ka toe ko te 0.

$$3x^{2}-52x=-48$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-52x}{3}=-\frac{48}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{52}{3}x=-\frac{48}{3}$$

Whakawehe -48 ki te 3.

$$x^{2}-\frac{52}{3}x=-16$$

Whakawehea te -\frac{52}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{26}{3}. Nā, tāpiria te pūrua o te -\frac{26}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{52}{3}x+\left(-\frac{26}{3}\right)^{2}=-16+\left(-\frac{26}{3}\right)^{2}$$

Pūruatia -\frac{26}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{52}{3}x+\frac{676}{9}=-16+\frac{676}{9}$$

Tāpiri -16 ki te \frac{676}{9}.

$$x^{2}-\frac{52}{3}x+\frac{676}{9}=\frac{532}{9}$$

Tauwehea x^{2}-\frac{52}{3}x+\frac{676}{9}. 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}.

$$\left(x-\frac{26}{3}\right)^{2}=\frac{532}{9}$$

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

$$\sqrt{\left(x-\frac{26}{3}\right)^{2}}=\sqrt{\frac{532}{9}}$$

Whakarūnātia.

$$x-\frac{26}{3}=\frac{2\sqrt{133}}{3} x-\frac{26}{3}=-\frac{2\sqrt{133}}{3}$$

Me tāpiri \frac{26}{3} ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{133}+26}{3} x=\frac{26-2\sqrt{133}}{3}$$