Whakaoti i te -8x^2+22x-15

$$- 8 x ^ { 2 } + 22 x - 15$$

Tauwehe

$$\left(5-4x\right)\left(2x-3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -8x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=22 ab=-8\left(-15\right)=120$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 120.

$$1,120 2,60 3,40 4,30 5,24 6,20 8,15 10,12$$

Tātaihia te tapeke mō ia takirua.

$$1+120=121 2+60=62 3+40=43 4+30=34 5+24=29 6+20=26 8+15=23 10+12=22$$

Ko te otinga te takirua ka hoatu i te tapeke 22.

$$a=12 b=10$$

Tuhia anō te -8x^{2}+22x-15 hei \left(-8x^{2}+12x\right)+\left(10x-15\right).

$$\left(-8x^{2}+12x\right)+\left(10x-15\right)$$

Tauwehea te -4x i te tuatahi me te 5 i te rōpū tuarua.

$$-4x\left(2x-3\right)+5\left(2x-3\right)$$

Whakatauwehea atu te kīanga pātahi 2x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-3\right)\left(-4x+5\right)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$-8x^{2}+22x-15=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\left(-8\right)\left(-15\right)}}{2\left(-8\right)}$$

Pūrua 22.

$$x=\frac{-22±\sqrt{484-4\left(-8\right)\left(-15\right)}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$x=\frac{-22±\sqrt{484+32\left(-15\right)}}{2\left(-8\right)}$$

Whakareatia 32 ki te -15.

$$x=\frac{-22±\sqrt{484-480}}{2\left(-8\right)}$$

Tāpiri 484 ki te -480.

$$x=\frac{-22±\sqrt{4}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 4.

$$x=\frac{-22±2}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$x=\frac{-22±2}{-16}$$

Nā, me whakaoti te whārite x=\frac{-22±2}{-16} ina he tāpiri te ±. Tāpiri -22 ki te 2.

$$x=-\frac{20}{-16}$$

Whakahekea te hautanga \frac{-20}{-16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{5}{4}$$

Nā, me whakaoti te whārite x=\frac{-22±2}{-16} ina he tango te ±. Tango 2 mai i -22.

$$x=-\frac{24}{-16}$$

Whakahekea te hautanga \frac{-24}{-16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=\frac{3}{2}$$

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te \frac{5}{4} mō te x_{1} me te \frac{3}{2} mō te x_{2}.

$$-8x^{2}+22x-15=-8\left(x-\frac{5}{4}\right)\left(x-\frac{3}{2}\right)$$

Tango \frac{5}{4} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$-8x^{2}+22x-15=-8\times \frac{-4x+5}{-4}\left(x-\frac{3}{2}\right)$$

Tango \frac{3}{2} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$-8x^{2}+22x-15=-8\times \frac{-4x+5}{-4}\times \frac{-2x+3}{-2}$$

Whakareatia \frac{-4x+5}{-4} ki te \frac{-2x+3}{-2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$-8x^{2}+22x-15=-8\times \frac{\left(-4x+5\right)\left(-2x+3\right)}{-4\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$-8x^{2}+22x-15=-8\times \frac{\left(-4x+5\right)\left(-2x+3\right)}{8}$$

Whakakorea atu te tauwehe pūnoa nui rawa 8 i roto i te -8 me te 8.

$$-8x^{2}+22x-15=-\left(-4x+5\right)\left(-2x+3\right)$$

Aromātai

$$\left(5-4x\right)\left(2x-3\right)$$