Whakaoti i te (9-x)(x-4)=3x

$$( 9 - x ) ( x - 4 ) = 3 x$$

Whakaoti mō x (complex solution)

$$x=-\sqrt{11}i+5\approx 5-3.31662479i$$

$$x=5+\sqrt{11}i\approx 5+3.31662479i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 9-x ki te x-4 ka whakakotahi i ngā kupu rite.

$$13x-36-x^{2}=3x$$

Tangohia te 3x mai i ngā taha e rua.

$$13x-36-x^{2}-3x=0$$

Pahekotia te 13x me -3x, ka 10x.

$$10x-36-x^{2}=0$$

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.

$$-x^{2}+10x-36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 10 mō b, me -36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-10±\sqrt{100+4\left(-36\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -36.

$$x=\frac{-10±\sqrt{100-144}}{2\left(-1\right)}$$

Tāpiri 100 ki te -144.

$$x=\frac{-10±\sqrt{-44}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -44.

$$x=\frac{-10±2\sqrt{11}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-10±2\sqrt{11}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{11}i}{-2} ina he tāpiri te ±. Tāpiri -10 ki te 2i\sqrt{11}.

$$x=\frac{-10+2\sqrt{11}i}{-2}$$

Whakawehe -10+2i\sqrt{11} ki te -2.

$$x=-\sqrt{11}i+5$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{11}i}{-2} ina he tango te ±. Tango 2i\sqrt{11} mai i -10.

$$x=\frac{-2\sqrt{11}i-10}{-2}$$

Whakawehe -10-2i\sqrt{11} ki te -2.

$$x=5+\sqrt{11}i$$

Kua oti te whārite te whakatau.

$$x=-\sqrt{11}i+5 x=5+\sqrt{11}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 9-x ki te x-4 ka whakakotahi i ngā kupu rite.

$$13x-36-x^{2}=3x$$

Tangohia te 3x mai i ngā taha e rua.

$$13x-36-x^{2}-3x=0$$

Pahekotia te 13x me -3x, ka 10x.

$$10x-36-x^{2}=0$$

Me tāpiri te 36 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$10x-x^{2}=36$$

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.

$$-x^{2}+10x=36$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+10x}{-1}=\frac{36}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{10}{-1}x=\frac{36}{-1}$$

Whakawehe 10 ki te -1.

$$x^{2}-10x=\frac{36}{-1}$$

Whakawehe 36 ki te -1.

$$x^{2}-10x=-36$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=-36+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=-36+25$$

Tāpiri -36 ki te 25.

$$x^{2}-10x+25=-11$$

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}.

$$\left(x-5\right)^{2}=-11$$

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

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{-11}$$

Whakarūnātia.

$$x-5=\sqrt{11}i x-5=-\sqrt{11}i$$

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

$$x=5+\sqrt{11}i x=-\sqrt{11}i+5$$