Whakaoti i te (20-5x)(6-x)=125*9

$$(20-5x)(6-x)=125 \times 9$$

Whakaoti mō x

$$x=\sqrt{226}+5\approx 20.033296378$$

$$x=5-\sqrt{226}\approx -10.033296378$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-5x ki te 6-x ka whakakotahi i ngā kupu rite.

$$120-50x+5x^{2}=125\times 9$$

Whakareatia te 125 ki te 9, ka 1125.

$$120-50x+5x^{2}=1125$$

Tangohia te 1125 mai i ngā taha e rua.

$$120-50x+5x^{2}-1125=0$$

Tangohia te 1125 i te 120, ka -1005.

$$-1005-50x+5x^{2}=0$$

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.

$$5x^{2}-50x-1005=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -50 mō b, me -1005 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 5\left(-1005\right)}}{2\times 5}$$

Pūrua -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 5\left(-1005\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-50\right)±\sqrt{2500-20\left(-1005\right)}}{2\times 5}$$

Whakareatia -20 ki te -1005.

$$x=\frac{-\left(-50\right)±\sqrt{2500+20100}}{2\times 5}$$

Tāpiri 2500 ki te 20100.

$$x=\frac{-\left(-50\right)±\sqrt{22600}}{2\times 5}$$

Tuhia te pūtakerua o te 22600.

$$x=\frac{-\left(-50\right)±10\sqrt{226}}{2\times 5}$$

Ko te tauaro o -50 ko 50.

$$x=\frac{50±10\sqrt{226}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{50±10\sqrt{226}}{10}$$

Nā, me whakaoti te whārite x=\frac{50±10\sqrt{226}}{10} ina he tāpiri te ±. Tāpiri 50 ki te 10\sqrt{226}.

$$x=\frac{10\sqrt{226}+50}{10}$$

Whakawehe 50+10\sqrt{226} ki te 10.

$$x=\sqrt{226}+5$$

Nā, me whakaoti te whārite x=\frac{50±10\sqrt{226}}{10} ina he tango te ±. Tango 10\sqrt{226} mai i 50.

$$x=\frac{50-10\sqrt{226}}{10}$$

Whakawehe 50-10\sqrt{226} ki te 10.

$$x=5-\sqrt{226}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{226}+5 x=5-\sqrt{226}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-5x ki te 6-x ka whakakotahi i ngā kupu rite.

$$120-50x+5x^{2}=125\times 9$$

Whakareatia te 125 ki te 9, ka 1125.

$$120-50x+5x^{2}=1125$$

Tangohia te 120 mai i ngā taha e rua.

$$-50x+5x^{2}=1125-120$$

Tangohia te 120 i te 1125, ka 1005.

$$-50x+5x^{2}=1005$$

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.

$$5x^{2}-50x=1005$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-50x}{5}=\frac{1005}{5}$$

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

$$x^{2}+\left(-\frac{50}{5}\right)x=\frac{1005}{5}$$

Whakawehe -50 ki te 5.

$$x^{2}-10x=\frac{1005}{5}$$

Whakawehe 1005 ki te 5.

$$x^{2}-10x=201$$

Whakawehea te -10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -5. Nā, tāpiria te pūrua o te -5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-10x+\left(-5\right)^{2}=201+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=201+25$$

Tāpiri 201 ki te 25.

$$x^{2}-10x+25=226$$

Tauwehea x^{2}-10x+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}.

$$\left(x-5\right)^{2}=226$$

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

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{226}$$

Whakarūnātia.

$$x-5=\sqrt{226} x-5=-\sqrt{226}$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$x=\sqrt{226}+5 x=5-\sqrt{226}$$