Whakaoti i te 90*(x-10)*(x-9)=1

$$90 \times (x-10) \times (x-9)=1$$

Whakaoti mō x

$$x=\frac{\sqrt{235}}{30}+\frac{19}{2}\approx 10.010990324$$

$$x=-\frac{\sqrt{235}}{30}+\frac{19}{2}\approx 8.989009676$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 90 ki te x-10.

$$\left(90x-900\right)\left(x-9\right)=1$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 90x-900 ki te x-9 ka whakakotahi i ngā kupu rite.

$$90x^{2}-1710x+8100=1$$

Tangohia te 1 mai i ngā taha e rua.

$$90x^{2}-1710x+8100-1=0$$

Tangohia te 1 i te 8100, ka 8099.

$$90x^{2}-1710x+8099=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 90 mō a, -1710 mō b, me 8099 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1710\right)±\sqrt{\left(-1710\right)^{2}-4\times 90\times 8099}}{2\times 90}$$

Pūrua -1710.

$$x=\frac{-\left(-1710\right)±\sqrt{2924100-4\times 90\times 8099}}{2\times 90}$$

Whakareatia -4 ki te 90.

$$x=\frac{-\left(-1710\right)±\sqrt{2924100-360\times 8099}}{2\times 90}$$

Whakareatia -360 ki te 8099.

$$x=\frac{-\left(-1710\right)±\sqrt{2924100-2915640}}{2\times 90}$$

Tāpiri 2924100 ki te -2915640.

$$x=\frac{-\left(-1710\right)±\sqrt{8460}}{2\times 90}$$

Tuhia te pūtakerua o te 8460.

$$x=\frac{-\left(-1710\right)±6\sqrt{235}}{2\times 90}$$

Ko te tauaro o -1710 ko 1710.

$$x=\frac{1710±6\sqrt{235}}{2\times 90}$$

Whakareatia 2 ki te 90.

$$x=\frac{1710±6\sqrt{235}}{180}$$

Nā, me whakaoti te whārite x=\frac{1710±6\sqrt{235}}{180} ina he tāpiri te ±. Tāpiri 1710 ki te 6\sqrt{235}.

$$x=\frac{6\sqrt{235}+1710}{180}$$

Whakawehe 1710+6\sqrt{235} ki te 180.

$$x=\frac{\sqrt{235}}{30}+\frac{19}{2}$$

Nā, me whakaoti te whārite x=\frac{1710±6\sqrt{235}}{180} ina he tango te ±. Tango 6\sqrt{235} mai i 1710.

$$x=\frac{1710-6\sqrt{235}}{180}$$

Whakawehe 1710-6\sqrt{235} ki te 180.

$$x=-\frac{\sqrt{235}}{30}+\frac{19}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{235}}{30}+\frac{19}{2} x=-\frac{\sqrt{235}}{30}+\frac{19}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 90 ki te x-10.

$$\left(90x-900\right)\left(x-9\right)=1$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 90x-900 ki te x-9 ka whakakotahi i ngā kupu rite.

$$90x^{2}-1710x+8100=1$$

Tangohia te 8100 mai i ngā taha e rua.

$$90x^{2}-1710x=1-8100$$

Tangohia te 8100 i te 1, ka -8099.

$$90x^{2}-1710x=-8099$$

Whakawehea ngā taha e rua ki te 90.

$$\frac{90x^{2}-1710x}{90}=-\frac{8099}{90}$$

Mā te whakawehe ki te 90 ka wetekia te whakareanga ki te 90.

$$x^{2}+\left(-\frac{1710}{90}\right)x=-\frac{8099}{90}$$

Whakawehe -1710 ki te 90.

$$x^{2}-19x=-\frac{8099}{90}$$

Whakawehea te -19, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{19}{2}. Nā, tāpiria te pūrua o te -\frac{19}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=-\frac{8099}{90}+\left(-\frac{19}{2}\right)^{2}$$

Pūruatia -\frac{19}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-19x+\frac{361}{4}=-\frac{8099}{90}+\frac{361}{4}$$

Tāpiri -\frac{8099}{90} ki te \frac{361}{4} 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.

$$x^{2}-19x+\frac{361}{4}=\frac{47}{180}$$

Tauwehea x^{2}-19x+\frac{361}{4}. 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}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{47}{180}$$

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

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{47}{180}}$$

Whakarūnātia.

$$x-\frac{19}{2}=\frac{\sqrt{235}}{30} x-\frac{19}{2}=-\frac{\sqrt{235}}{30}$$

Me tāpiri \frac{19}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{235}}{30}+\frac{19}{2} x=-\frac{\sqrt{235}}{30}+\frac{19}{2}$$