Whakaoti i te (60-40-x)(100+10x)=2240

$$( 60 - 40 - x ) ( 100 + 10 x ) = 2240$$

Whakaoti mō x

$$x=4$$

$$x=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 40 i te 60, ka 20.

$$\left(20-x\right)\left(100+10x\right)=2240$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-x ki te 100+10x ka whakakotahi i ngā kupu rite.

$$2000+100x-10x^{2}=2240$$

Tangohia te 2240 mai i ngā taha e rua.

$$2000+100x-10x^{2}-2240=0$$

Tangohia te 2240 i te 2000, ka -240.

$$-240+100x-10x^{2}=0$$

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.

$$-10x^{2}+100x-240=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -10 mō a, 100 mō b, me -240 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-10\right)\left(-240\right)}}{2\left(-10\right)}$$

Pūrua 100.

$$x=\frac{-100±\sqrt{10000-4\left(-10\right)\left(-240\right)}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$x=\frac{-100±\sqrt{10000+40\left(-240\right)}}{2\left(-10\right)}$$

Whakareatia 40 ki te -240.

$$x=\frac{-100±\sqrt{10000-9600}}{2\left(-10\right)}$$

Tāpiri 10000 ki te -9600.

$$x=\frac{-100±\sqrt{400}}{2\left(-10\right)}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-100±20}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$x=\frac{-100±20}{-20}$$

Nā, me whakaoti te whārite x=\frac{-100±20}{-20} ina he tāpiri te ±. Tāpiri -100 ki te 20.

$$x=-\frac{80}{-20}$$

Whakawehe -80 ki te -20.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-100±20}{-20} ina he tango te ±. Tango 20 mai i -100.

$$x=-\frac{120}{-20}$$

Whakawehe -120 ki te -20.

$$x=6$$

Kua oti te whārite te whakatau.

$$x=4 x=6$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 40 i te 60, ka 20.

$$\left(20-x\right)\left(100+10x\right)=2240$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-x ki te 100+10x ka whakakotahi i ngā kupu rite.

$$2000+100x-10x^{2}=2240$$

Tangohia te 2000 mai i ngā taha e rua.

$$100x-10x^{2}=2240-2000$$

Tangohia te 2000 i te 2240, ka 240.

$$100x-10x^{2}=240$$

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.

$$-10x^{2}+100x=240$$

Whakawehea ngā taha e rua ki te -10.

$$\frac{-10x^{2}+100x}{-10}=\frac{240}{-10}$$

Mā te whakawehe ki te -10 ka wetekia te whakareanga ki te -10.

$$x^{2}+\frac{100}{-10}x=\frac{240}{-10}$$

Whakawehe 100 ki te -10.

$$x^{2}-10x=\frac{240}{-10}$$

Whakawehe 240 ki te -10.

$$x^{2}-10x=-24$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=-24+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=-24+25$$

Tāpiri -24 ki te 25.

$$x^{2}-10x+25=1$$

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}.

$$\left(x-5\right)^{2}=1$$

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

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x-5=1 x-5=-1$$

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

$$x=6 x=4$$