Whakaoti i te left(40-xright)(30-x)=1140

$$\left( 40-x \right) (30-x) = 1140$$

Whakaoti mō x

$$x=35-\sqrt{1165}\approx 0.867903668$$

$$x=\sqrt{1165}+35\approx 69.132096332$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-x ki te 30-x ka whakakotahi i ngā kupu rite.

$$1200-70x+x^{2}=1140$$

Tangohia te 1140 mai i ngā taha e rua.

$$1200-70x+x^{2}-1140=0$$

Tangohia te 1140 i te 1200, ka 60.

$$60-70x+x^{2}=0$$

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.

$$x^{2}-70x+60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -70 mō b, me 60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 60}}{2}$$

Pūrua -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 60}}{2}$$

Whakareatia -4 ki te 60.

$$x=\frac{-\left(-70\right)±\sqrt{4900-240}}{2}$$

Tāpiri 4900 ki te -240.

$$x=\frac{-\left(-70\right)±\sqrt{4660}}{2}$$

Tuhia te pūtakerua o te 4660.

$$x=\frac{-\left(-70\right)±2\sqrt{1165}}{2}$$

Ko te tauaro o -70 ko 70.

$$x=\frac{70±2\sqrt{1165}}{2}$$

Nā, me whakaoti te whārite x=\frac{70±2\sqrt{1165}}{2} ina he tāpiri te ±. Tāpiri 70 ki te 2\sqrt{1165}.

$$x=\frac{2\sqrt{1165}+70}{2}$$

Whakawehe 70+2\sqrt{1165} ki te 2.

$$x=\sqrt{1165}+35$$

Nā, me whakaoti te whārite x=\frac{70±2\sqrt{1165}}{2} ina he tango te ±. Tango 2\sqrt{1165} mai i 70.

$$x=\frac{70-2\sqrt{1165}}{2}$$

Whakawehe 70-2\sqrt{1165} ki te 2.

$$x=35-\sqrt{1165}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{1165}+35 x=35-\sqrt{1165}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-x ki te 30-x ka whakakotahi i ngā kupu rite.

$$1200-70x+x^{2}=1140$$

Tangohia te 1200 mai i ngā taha e rua.

$$-70x+x^{2}=1140-1200$$

Tangohia te 1200 i te 1140, ka -60.

$$-70x+x^{2}=-60$$

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.

$$x^{2}-70x=-60$$

Whakawehea te -70, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -35. Nā, tāpiria te pūrua o te -35 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-70x+\left(-35\right)^{2}=-60+\left(-35\right)^{2}$$

Pūrua -35.

$$x^{2}-70x+1225=-60+1225$$

Tāpiri -60 ki te 1225.

$$x^{2}-70x+1225=1165$$

Tauwehea x^{2}-70x+1225. 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}.

$$\left(x-35\right)^{2}=1165$$

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

$$\sqrt{\left(x-35\right)^{2}}=\sqrt{1165}$$

Whakarūnātia.

$$x-35=\sqrt{1165} x-35=-\sqrt{1165}$$

Me tāpiri 35 ki ngā taha e rua o te whārite.

$$x=\sqrt{1165}+35 x=35-\sqrt{1165}$$