Whakaoti i te 16(x-4)-(x+3)=(x+3)(x-4)

$$16 ( x - 4 ) - ( x + 3 ) = ( x + 3 ) ( x - 4 )$$

Whakaoti mō x

$$x=5$$

$$x=11$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te x-4.

$$16x-64-\left(x+3\right)=\left(x+3\right)\left(x-4\right)$$

Hei kimi i te tauaro o x+3, kimihia te tauaro o ia taurangi.

$$16x-64-x-3=\left(x+3\right)\left(x-4\right)$$

Pahekotia te 16x me -x, ka 15x.

$$15x-64-3=\left(x+3\right)\left(x-4\right)$$

Tangohia te 3 i te -64, ka -67.

$$15x-67=\left(x+3\right)\left(x-4\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te x-4 ka whakakotahi i ngā kupu rite.

$$15x-67=x^{2}-x-12$$

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

$$15x-67-x^{2}=-x-12$$

Me tāpiri te x ki ngā taha e rua.

$$15x-67-x^{2}+x=-12$$

Pahekotia te 15x me x, ka 16x.

$$16x-67-x^{2}=-12$$

Me tāpiri te 12 ki ngā taha e rua.

$$16x-67-x^{2}+12=0$$

Tāpirihia te -67 ki te 12, ka -55.

$$16x-55-x^{2}=0$$

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.

$$-x^{2}+16x-55=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 16 mō b, me -55 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\left(-55\right)}}{2\left(-1\right)}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\left(-1\right)\left(-55\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-16±\sqrt{256+4\left(-55\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -55.

$$x=\frac{-16±\sqrt{256-220}}{2\left(-1\right)}$$

Tāpiri 256 ki te -220.

$$x=\frac{-16±\sqrt{36}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-16±6}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-16±6}{-2}$$

Nā, me whakaoti te whārite x=\frac{-16±6}{-2} ina he tāpiri te ±. Tāpiri -16 ki te 6.

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

Whakawehe -10 ki te -2.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{-16±6}{-2} ina he tango te ±. Tango 6 mai i -16.

$$x=-\frac{22}{-2}$$

Whakawehe -22 ki te -2.

$$x=11$$

Kua oti te whārite te whakatau.

$$x=5 x=11$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te x-4.

$$16x-64-\left(x+3\right)=\left(x+3\right)\left(x-4\right)$$

Hei kimi i te tauaro o x+3, kimihia te tauaro o ia taurangi.

$$16x-64-x-3=\left(x+3\right)\left(x-4\right)$$

Pahekotia te 16x me -x, ka 15x.

$$15x-64-3=\left(x+3\right)\left(x-4\right)$$

Tangohia te 3 i te -64, ka -67.

$$15x-67=\left(x+3\right)\left(x-4\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te x-4 ka whakakotahi i ngā kupu rite.

$$15x-67=x^{2}-x-12$$

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

$$15x-67-x^{2}=-x-12$$

Me tāpiri te x ki ngā taha e rua.

$$15x-67-x^{2}+x=-12$$

Pahekotia te 15x me x, ka 16x.

$$16x-67-x^{2}=-12$$

Me tāpiri te 67 ki ngā taha e rua.

$$16x-x^{2}=-12+67$$

Tāpirihia te -12 ki te 67, ka 55.

$$16x-x^{2}=55$$

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.

$$-x^{2}+16x=55$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+16x}{-1}=\frac{55}{-1}$$

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

$$x^{2}+\frac{16}{-1}x=\frac{55}{-1}$$

Whakawehe 16 ki te -1.

$$x^{2}-16x=\frac{55}{-1}$$

Whakawehe 55 ki te -1.

$$x^{2}-16x=-55$$

Whakawehea te -16, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -8. Nā, tāpiria te pūrua o te -8 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-16x+\left(-8\right)^{2}=-55+\left(-8\right)^{2}$$

Pūrua -8.

$$x^{2}-16x+64=-55+64$$

Tāpiri -55 ki te 64.

$$x^{2}-16x+64=9$$

Tauwehea x^{2}-16x+64. 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}.

$$\left(x-8\right)^{2}=9$$

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

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x-8=3 x-8=-3$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$x=11 x=5$$