Whakaoti i te -2x^2+20x=48

$$-2 { x }^{ 2 } +20x=48$$

Whakaoti mō x

$$x=4$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 48 mai i ngā taha e rua.

$$-2x^{2}+20x-48=0$$

Whakawehea ngā taha e rua ki te 2.

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

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.

$$a+b=10 ab=-\left(-24\right)=24$$

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 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

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.

$$-2x^{2}+20x=48$$

Me tango 48 mai i ngā taha e rua o te whārite.

$$-2x^{2}+20x-48=48-48$$

Mā te tango i te 48 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+20x-48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 20 mō b, me -48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-2\right)\left(-48\right)}}{2\left(-2\right)}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\left(-2\right)\left(-48\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-20±\sqrt{400+8\left(-48\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -48.

$$x=\frac{-20±\sqrt{400-384}}{2\left(-2\right)}$$

Tāpiri 400 ki te -384.

$$x=\frac{-20±\sqrt{16}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-20±4}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-20±4}{-4}$$

Nā, me whakaoti te whārite x=\frac{-20±4}{-4} ina he tāpiri te ±. Tāpiri -20 ki te 4.

$$x=-\frac{16}{-4}$$

Whakawehe -16 ki te -4.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-20±4}{-4} ina he tango te ±. Tango 4 mai i -20.

$$x=-\frac{24}{-4}$$

Whakawehe -24 ki te -4.

$$x=6$$

Kua oti te whārite te whakatau.

$$x=4 x=6$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$-2x^{2}+20x=48$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+20x}{-2}=\frac{48}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$x^{2}+\frac{20}{-2}x=\frac{48}{-2}$$

Whakawehe 20 ki te -2.

$$x^{2}-10x=\frac{48}{-2}$$

Whakawehe 48 ki te -2.

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

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 x^{2}-10x+25. 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}.

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

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.

$$x=6 x=4$$