Whakaoti i te (76-4x)(8+2x)=1080

$$( 76 - 4 x ) ( 8 + 2 x ) = 1080$$

Whakaoti mō x (complex solution)

$$x=\frac{15+\sqrt{11}i}{2}\approx 7.5+1.658312395i$$

$$x=\frac{-\sqrt{11}i+15}{2}\approx 7.5-1.658312395i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 76-4x ki te 8+2x ka whakakotahi i ngā kupu rite.

$$608+120x-8x^{2}=1080$$

Tangohia te 1080 mai i ngā taha e rua.

$$608+120x-8x^{2}-1080=0$$

Tangohia te 1080 i te 608, ka -472.

$$-472+120x-8x^{2}=0$$

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.

$$-8x^{2}+120x-472=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -8 mō a, 120 mō b, me -472 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-8\right)\left(-472\right)}}{2\left(-8\right)}$$

Pūrua 120.

$$x=\frac{-120±\sqrt{14400-4\left(-8\right)\left(-472\right)}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$x=\frac{-120±\sqrt{14400+32\left(-472\right)}}{2\left(-8\right)}$$

Whakareatia 32 ki te -472.

$$x=\frac{-120±\sqrt{14400-15104}}{2\left(-8\right)}$$

Tāpiri 14400 ki te -15104.

$$x=\frac{-120±\sqrt{-704}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te -704.

$$x=\frac{-120±8\sqrt{11}i}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$x=\frac{-120±8\sqrt{11}i}{-16}$$

Nā, me whakaoti te whārite x=\frac{-120±8\sqrt{11}i}{-16} ina he tāpiri te ±. Tāpiri -120 ki te 8i\sqrt{11}.

$$x=\frac{-120+8\sqrt{11}i}{-16}$$

Whakawehe -120+8i\sqrt{11} ki te -16.

$$x=\frac{-\sqrt{11}i+15}{2}$$

Nā, me whakaoti te whārite x=\frac{-120±8\sqrt{11}i}{-16} ina he tango te ±. Tango 8i\sqrt{11} mai i -120.

$$x=\frac{-8\sqrt{11}i-120}{-16}$$

Whakawehe -120-8i\sqrt{11} ki te -16.

$$x=\frac{15+\sqrt{11}i}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{11}i+15}{2} x=\frac{15+\sqrt{11}i}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 76-4x ki te 8+2x ka whakakotahi i ngā kupu rite.

$$608+120x-8x^{2}=1080$$

Tangohia te 608 mai i ngā taha e rua.

$$120x-8x^{2}=1080-608$$

Tangohia te 608 i te 1080, ka 472.

$$120x-8x^{2}=472$$

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.

$$-8x^{2}+120x=472$$

Whakawehea ngā taha e rua ki te -8.

$$\frac{-8x^{2}+120x}{-8}=\frac{472}{-8}$$

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

$$x^{2}+\frac{120}{-8}x=\frac{472}{-8}$$

Whakawehe 120 ki te -8.

$$x^{2}-15x=\frac{472}{-8}$$

Whakawehe 472 ki te -8.

$$x^{2}-15x=-59$$

Whakawehea te -15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{2}. Nā, tāpiria te pūrua o te -\frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-59+\left(-\frac{15}{2}\right)^{2}$$

Pūruatia -\frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-15x+\frac{225}{4}=-59+\frac{225}{4}$$

Tāpiri -59 ki te \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=-\frac{11}{4}$$

Tauwehea x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=-\frac{11}{4}$$

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

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{-\frac{11}{4}}$$

Whakarūnātia.

$$x-\frac{15}{2}=\frac{\sqrt{11}i}{2} x-\frac{15}{2}=-\frac{\sqrt{11}i}{2}$$

Me tāpiri \frac{15}{2} ki ngā taha e rua o te whārite.

$$x=\frac{15+\sqrt{11}i}{2} x=\frac{-\sqrt{11}i+15}{2}$$