Whakaoti i te 5x^2-48x+20=0

$$5 { x }^{ 2 } -48x+20=0$$

Whakaoti mō x

$$x = \frac{2 \sqrt{119} + 24}{5} \approx 9.163484846$$

$$x=\frac{24-2\sqrt{119}}{5}\approx 0.436515154$$

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.

$$5x^{2}-48x+20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -48 mō b, me 20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 5\times 20}}{2\times 5}$$

Pūrua -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 5\times 20}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-48\right)±\sqrt{2304-20\times 20}}{2\times 5}$$

Whakareatia -20 ki te 20.

$$x=\frac{-\left(-48\right)±\sqrt{2304-400}}{2\times 5}$$

Tāpiri 2304 ki te -400.

$$x=\frac{-\left(-48\right)±\sqrt{1904}}{2\times 5}$$

Tuhia te pūtakerua o te 1904.

$$x=\frac{-\left(-48\right)±4\sqrt{119}}{2\times 5}$$

Ko te tauaro o -48 ko 48.

$$x=\frac{48±4\sqrt{119}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{48±4\sqrt{119}}{10}$$

Nā, me whakaoti te whārite x=\frac{48±4\sqrt{119}}{10} ina he tāpiri te ±. Tāpiri 48 ki te 4\sqrt{119}.

$$x=\frac{4\sqrt{119}+48}{10}$$

Whakawehe 48+4\sqrt{119} ki te 10.

$$x=\frac{2\sqrt{119}+24}{5}$$

Nā, me whakaoti te whārite x=\frac{48±4\sqrt{119}}{10} ina he tango te ±. Tango 4\sqrt{119} mai i 48.

$$x=\frac{48-4\sqrt{119}}{10}$$

Whakawehe 48-4\sqrt{119} ki te 10.

$$x=\frac{24-2\sqrt{119}}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{119}+24}{5} x=\frac{24-2\sqrt{119}}{5}$$

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.

$$5x^{2}-48x+20=0$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$5x^{2}-48x+20-20=-20$$

Mā te tango i te 20 i a ia ake anō ka toe ko te 0.

$$5x^{2}-48x=-20$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-48x}{5}=-\frac{20}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}-\frac{48}{5}x=-\frac{20}{5}$$

Whakawehe -20 ki te 5.

$$x^{2}-\frac{48}{5}x=-4$$

Whakawehea te -\frac{48}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{24}{5}. Nā, tāpiria te pūrua o te -\frac{24}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{48}{5}x+\left(-\frac{24}{5}\right)^{2}=-4+\left(-\frac{24}{5}\right)^{2}$$

Pūruatia -\frac{24}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=-4+\frac{576}{25}$$

Tāpiri -4 ki te \frac{576}{25}.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=\frac{476}{25}$$

Tauwehea x^{2}-\frac{48}{5}x+\frac{576}{25}. 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}.

$$\left(x-\frac{24}{5}\right)^{2}=\frac{476}{25}$$

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

$$\sqrt{\left(x-\frac{24}{5}\right)^{2}}=\sqrt{\frac{476}{25}}$$

Whakarūnātia.

$$x-\frac{24}{5}=\frac{2\sqrt{119}}{5} x-\frac{24}{5}=-\frac{2\sqrt{119}}{5}$$

Me tāpiri \frac{24}{5} ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{119}+24}{5} x=\frac{24-2\sqrt{119}}{5}$$