Whakaoti i te x^2-12x+19=-2x-5

$$x ^ { 2 } - 12 x + 19 = - 2 x - 5$$

Whakaoti mō x

$$x=4$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 2x ki ngā taha e rua.

$$x^{2}-12x+19+2x=-5$$

Pahekotia te -12x me 2x, ka -10x.

$$x^{2}-10x+19=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$x^{2}-10x+19+5=0$$

Tāpirihia te 19 ki te 5, ka 24.

$$x^{2}-10x+24=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-10x+24 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=24$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

Ko te otinga te takirua ka hoatu i te tapeke -10.

$$a=-6 b=-4$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-6\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x-4=0.

$$x=6 x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 2x ki ngā taha e rua.

$$x^{2}-12x+19+2x=-5$$

Pahekotia te -12x me 2x, ka -10x.

$$x^{2}-10x+19=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$x^{2}-10x+19+5=0$$

Tāpirihia te 19 ki te 5, ka 24.

$$x^{2}-10x+24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=1\times 24=24$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

Ko te otinga te takirua ka hoatu i te tapeke -10.

$$a=-6 b=-4$$

Tuhia anō te x^{2}-10x+24 hei \left(x^{2}-6x\right)+\left(-4x+24\right).

$$\left(x^{2}-6x\right)+\left(-4x+24\right)$$

Tauwehea te x i te tuatahi me te -4 i te rōpū tuarua.

$$x\left(x-6\right)-4\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x-4=0.

$$x=6 x=4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 2x ki ngā taha e rua.

$$x^{2}-12x+19+2x=-5$$

Pahekotia te -12x me 2x, ka -10x.

$$x^{2}-10x+19=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$x^{2}-10x+19+5=0$$

Tāpirihia te 19 ki te 5, ka 24.

$$x^{2}-10x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -10 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 24}}{2}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 24}}{2}$$

Whakareatia -4 ki te 24.

$$x=\frac{-\left(-10\right)±\sqrt{100-96}}{2}$$

Tāpiri 100 ki te -96.

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

Tuhia te pūtakerua o te 4.

$$x=\frac{-\left(-10\right)±2}{2}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±2}{2}$$

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

$$x=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$x=6$$

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

$$x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=6 x=4$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 2x ki ngā taha e rua.

$$x^{2}-12x+19+2x=-5$$

Pahekotia te -12x me 2x, ka -10x.

$$x^{2}-10x+19=-5$$

Tangohia te 19 mai i ngā taha e rua.

$$x^{2}-10x=-5-19$$

Tangohia te 19 i te -5, ka -24.

$$x^{2}-10x=-24$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=-24+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=-24+25$$

Tāpiri -24 ki te 25.

$$x^{2}-10x+25=1$$

Tauwehea te x^{2}-10x+25. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=1$$

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

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x-5=1 x-5=-1$$

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

$$x=6 x=4$$