Whakaoti i te (76-4x)(10+2x)=1080

$$(76-4x)(10+2x)=1080$$

Whakaoti mō x

$$x=4$$

$$x=10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 76-4x ki te 10+2x ka whakakotahi i ngā kupu rite.

$$760+112x-8x^{2}=1080$$

Tangohia te 1080 mai i ngā taha e rua.

$$760+112x-8x^{2}-1080=0$$

Tangohia te 1080 i te 760, ka -320.

$$-320+112x-8x^{2}=0$$

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.

$$-8x^{2}+112x-320=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -8 mō a, 112 mō b, me -320 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-112±\sqrt{112^{2}-4\left(-8\right)\left(-320\right)}}{2\left(-8\right)}$$

Pūrua 112.

$$x=\frac{-112±\sqrt{12544-4\left(-8\right)\left(-320\right)}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$x=\frac{-112±\sqrt{12544+32\left(-320\right)}}{2\left(-8\right)}$$

Whakareatia 32 ki te -320.

$$x=\frac{-112±\sqrt{12544-10240}}{2\left(-8\right)}$$

Tāpiri 12544 ki te -10240.

$$x=\frac{-112±\sqrt{2304}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 2304.

$$x=\frac{-112±48}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$x=\frac{-112±48}{-16}$$

Nā, me whakaoti te whārite x=\frac{-112±48}{-16} ina he tāpiri te ±. Tāpiri -112 ki te 48.

$$x=-\frac{64}{-16}$$

Whakawehe -64 ki te -16.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-112±48}{-16} ina he tango te ±. Tango 48 mai i -112.

$$x=-\frac{160}{-16}$$

Whakawehe -160 ki te -16.

$$x=10$$

Kua oti te whārite te whakatau.

$$x=4 x=10$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 76-4x ki te 10+2x ka whakakotahi i ngā kupu rite.

$$760+112x-8x^{2}=1080$$

Tangohia te 760 mai i ngā taha e rua.

$$112x-8x^{2}=1080-760$$

Tangohia te 760 i te 1080, ka 320.

$$112x-8x^{2}=320$$

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.

$$-8x^{2}+112x=320$$

Whakawehea ngā taha e rua ki te -8.

$$\frac{-8x^{2}+112x}{-8}=\frac{320}{-8}$$

Mā te whakawehe ki te -8 ka wetekia te whakareanga ki te -8.

$$x^{2}+\frac{112}{-8}x=\frac{320}{-8}$$

Whakawehe 112 ki te -8.

$$x^{2}-14x=\frac{320}{-8}$$

Whakawehe 320 ki te -8.

$$x^{2}-14x=-40$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-40+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-40+49$$

Tāpiri -40 ki te 49.

$$x^{2}-14x+49=9$$

Tauwehea x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=9$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x-7=3 x-7=-3$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=10 x=4$$