Whakaoti i te (4x-8)(x+5)=(5x-2)(x-2)

$$( 4 x - 8 ) ( x + 5 ) = ( 5 x - 2 ) ( x - 2 )$$

Whakaoti mō x

$$x=22$$

$$x=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 4x-8 ki te x+5 ka whakakotahi i ngā kupu rite.

$$4x^{2}+12x-40=\left(5x-2\right)\left(x-2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x-2 ki te x-2 ka whakakotahi i ngā kupu rite.

$$4x^{2}+12x-40=5x^{2}-12x+4$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$4x^{2}+12x-40-5x^{2}=-12x+4$$

Pahekotia te 4x^{2} me -5x^{2}, ka -x^{2}.

$$-x^{2}+12x-40=-12x+4$$

Me tāpiri te 12x ki ngā taha e rua.

$$-x^{2}+12x-40+12x=4$$

Pahekotia te 12x me 12x, ka 24x.

$$-x^{2}+24x-40=4$$

Tangohia te 4 mai i ngā taha e rua.

$$-x^{2}+24x-40-4=0$$

Tangohia te 4 i te -40, ka -44.

$$-x^{2}+24x-44=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 24 mō b, me -44 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\left(-44\right)}}{2\left(-1\right)}$$

Pūrua 24.

$$x=\frac{-24±\sqrt{576-4\left(-1\right)\left(-44\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-24±\sqrt{576+4\left(-44\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -44.

$$x=\frac{-24±\sqrt{576-176}}{2\left(-1\right)}$$

Tāpiri 576 ki te -176.

$$x=\frac{-24±\sqrt{400}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-24±20}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-24±20}{-2}$$

Nā, me whakaoti te whārite x=\frac{-24±20}{-2} ina he tāpiri te ±. Tāpiri -24 ki te 20.

$$x=-\frac{4}{-2}$$

Whakawehe -4 ki te -2.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{-24±20}{-2} ina he tango te ±. Tango 20 mai i -24.

$$x=-\frac{44}{-2}$$

Whakawehe -44 ki te -2.

$$x=22$$

Kua oti te whārite te whakatau.

$$x=2 x=22$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 4x-8 ki te x+5 ka whakakotahi i ngā kupu rite.

$$4x^{2}+12x-40=\left(5x-2\right)\left(x-2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x-2 ki te x-2 ka whakakotahi i ngā kupu rite.

$$4x^{2}+12x-40=5x^{2}-12x+4$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$4x^{2}+12x-40-5x^{2}=-12x+4$$

Pahekotia te 4x^{2} me -5x^{2}, ka -x^{2}.

$$-x^{2}+12x-40=-12x+4$$

Me tāpiri te 12x ki ngā taha e rua.

$$-x^{2}+12x-40+12x=4$$

Pahekotia te 12x me 12x, ka 24x.

$$-x^{2}+24x-40=4$$

Me tāpiri te 40 ki ngā taha e rua.

$$-x^{2}+24x=4+40$$

Tāpirihia te 4 ki te 40, ka 44.

$$-x^{2}+24x=44$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+24x}{-1}=\frac{44}{-1}$$

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

$$x^{2}+\frac{24}{-1}x=\frac{44}{-1}$$

Whakawehe 24 ki te -1.

$$x^{2}-24x=\frac{44}{-1}$$

Whakawehe 44 ki te -1.

$$x^{2}-24x=-44$$

Whakawehea te -24, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -12. Nā, tāpiria te pūrua o te -12 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-24x+\left(-12\right)^{2}=-44+\left(-12\right)^{2}$$

Pūrua -12.

$$x^{2}-24x+144=-44+144$$

Tāpiri -44 ki te 144.

$$x^{2}-24x+144=100$$

Tauwehea x^{2}-24x+144. 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}.

$$\left(x-12\right)^{2}=100$$

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

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{100}$$

Whakarūnātia.

$$x-12=10 x-12=-10$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$x=22 x=2$$