Whakaoti i te -k+3/4-k=k-3

$$\frac { - k + 3 } { 4 - k } = k - 3$$

Whakaoti mō k

$$k=3$$

$$k=5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe k ki 4 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te -k+4.

$$-k+3=\left(-k+4\right)k+\left(-k+4\right)\left(-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -k+4 ki te k.

$$-k+3=-k^{2}+4k+\left(-k+4\right)\left(-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -k+4 ki te -3.

$$-k+3=-k^{2}+4k+3k-12$$

Pahekotia te 4k me 3k, ka 7k.

$$-k+3=-k^{2}+7k-12$$

Me tāpiri te k^{2} ki ngā taha e rua.

$$-k+3+k^{2}=7k-12$$

Tangohia te 7k mai i ngā taha e rua.

$$-k+3+k^{2}-7k=-12$$

Me tāpiri te 12 ki ngā taha e rua.

$$-k+3+k^{2}-7k+12=0$$

Tāpirihia te 3 ki te 12, ka 15.

$$-k+15+k^{2}-7k=0$$

Pahekotia te -k me -7k, ka -8k.

$$-8k+15+k^{2}=0$$

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.

$$k^{2}-8k+15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me 15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 15}}{2}$$

Pūrua -8.

$$k=\frac{-\left(-8\right)±\sqrt{64-4\times 15}}{2}$$

Whakareatia -4 ki te 15.

$$k=\frac{-\left(-8\right)±\sqrt{64-60}}{2}$$

Tāpiri 64 ki te -60.

$$k=\frac{-\left(-8\right)±\sqrt{4}}{2}$$

Tuhia te pūtakerua o te 4.

$$k=\frac{-\left(-8\right)±2}{2}$$

Ko te tauaro o -8 ko 8.

$$k=\frac{8±2}{2}$$

Nā, me whakaoti te whārite k=\frac{8±2}{2} ina he tāpiri te ±. Tāpiri 8 ki te 2.

$$k=\frac{10}{2}$$

Whakawehe 10 ki te 2.

$$k=5$$

Nā, me whakaoti te whārite k=\frac{8±2}{2} ina he tango te ±. Tango 2 mai i 8.

$$k=\frac{6}{2}$$

Whakawehe 6 ki te 2.

$$k=3$$

Kua oti te whārite te whakatau.

$$k=5 k=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe k ki 4 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te -k+4.

$$-k+3=\left(-k+4\right)k+\left(-k+4\right)\left(-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -k+4 ki te k.

$$-k+3=-k^{2}+4k+\left(-k+4\right)\left(-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -k+4 ki te -3.

$$-k+3=-k^{2}+4k+3k-12$$

Pahekotia te 4k me 3k, ka 7k.

$$-k+3=-k^{2}+7k-12$$

Me tāpiri te k^{2} ki ngā taha e rua.

$$-k+3+k^{2}=7k-12$$

Tangohia te 7k mai i ngā taha e rua.

$$-k+3+k^{2}-7k=-12$$

Tangohia te 3 mai i ngā taha e rua.

$$-k+k^{2}-7k=-12-3$$

Tangohia te 3 i te -12, ka -15.

$$-k+k^{2}-7k=-15$$

Pahekotia te -k me -7k, ka -8k.

$$-8k+k^{2}=-15$$

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.

$$k^{2}-8k=-15$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}-8k+\left(-4\right)^{2}=-15+\left(-4\right)^{2}$$

Pūrua -4.

$$k^{2}-8k+16=-15+16$$

Tāpiri -15 ki te 16.

$$k^{2}-8k+16=1$$

Tauwehea k^{2}-8k+16. 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}.

$$\left(k-4\right)^{2}=1$$

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

$$\sqrt{\left(k-4\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$k-4=1 k-4=-1$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$k=5 k=3$$