Whakaoti i te 60x^2+169x+60=0

$$60 x ^ { 2 } + 169 x + 60 = 0$$

Whakaoti mō x

$$x = -\frac{12}{5} = -2\frac{2}{5} = -2.4$$

$$x=-\frac{5}{12}\approx -0.416666667$$

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.

$$60x^{2}+169x+60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 60 mō a, 169 mō b, me 60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-169±\sqrt{169^{2}-4\times 60\times 60}}{2\times 60}$$

Pūrua 169.

$$x=\frac{-169±\sqrt{28561-4\times 60\times 60}}{2\times 60}$$

Whakareatia -4 ki te 60.

$$x=\frac{-169±\sqrt{28561-240\times 60}}{2\times 60}$$

Whakareatia -240 ki te 60.

$$x=\frac{-169±\sqrt{28561-14400}}{2\times 60}$$

Tāpiri 28561 ki te -14400.

$$x=\frac{-169±\sqrt{14161}}{2\times 60}$$

Tuhia te pūtakerua o te 14161.

$$x=\frac{-169±119}{2\times 60}$$

Whakareatia 2 ki te 60.

$$x=\frac{-169±119}{120}$$

Nā, me whakaoti te whārite x=\frac{-169±119}{120} ina he tāpiri te ±. Tāpiri -169 ki te 119.

$$x=-\frac{50}{120}$$

Whakahekea te hautanga \frac{-50}{120} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=-\frac{5}{12}$$

Nā, me whakaoti te whārite x=\frac{-169±119}{120} ina he tango te ±. Tango 119 mai i -169.

$$x=-\frac{288}{120}$$

Whakahekea te hautanga \frac{-288}{120} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 24.

$$x=-\frac{12}{5}$$

Kua oti te whārite te whakatau.

$$x=-\frac{5}{12} x=-\frac{12}{5}$$

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.

$$60x^{2}+169x+60=0$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$60x^{2}+169x+60-60=-60$$

Mā te tango i te 60 i a ia ake anō ka toe ko te 0.

$$60x^{2}+169x=-60$$

Whakawehea ngā taha e rua ki te 60.

$$\frac{60x^{2}+169x}{60}=-\frac{60}{60}$$

Mā te whakawehe ki te 60 ka wetekia te whakareanga ki te 60.

$$x^{2}+\frac{169}{60}x=-\frac{60}{60}$$

Whakawehe -60 ki te 60.

$$x^{2}+\frac{169}{60}x=-1$$

Whakawehea te \frac{169}{60}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{169}{120}. Nā, tāpiria te pūrua o te \frac{169}{120} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{169}{60}x+\left(\frac{169}{120}\right)^{2}=-1+\left(\frac{169}{120}\right)^{2}$$

Pūruatia \frac{169}{120} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{169}{60}x+\frac{28561}{14400}=-1+\frac{28561}{14400}$$

Tāpiri -1 ki te \frac{28561}{14400}.

$$x^{2}+\frac{169}{60}x+\frac{28561}{14400}=\frac{14161}{14400}$$

Tauwehea x^{2}+\frac{169}{60}x+\frac{28561}{14400}. 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}.

$$\left(x+\frac{169}{120}\right)^{2}=\frac{14161}{14400}$$

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

$$\sqrt{\left(x+\frac{169}{120}\right)^{2}}=\sqrt{\frac{14161}{14400}}$$

Whakarūnātia.

$$x+\frac{169}{120}=\frac{119}{120} x+\frac{169}{120}=-\frac{119}{120}$$

Me tango \frac{169}{120} mai i ngā taha e rua o te whārite.

$$x=-\frac{5}{12} x=-\frac{12}{5}$$