Whakaoti i te x^2+20x+75=0

$$x ^ { 2 } + 20 x + 75 = 0$$

Whakaoti mō x

$$x=-15$$

$$x=-5$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+20x+75 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.

$$a+b=20 ab=75$$

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 75.

$$1,75 3,25 5,15$$

Tātaihia te tapeke mō ia takirua.

$$1+75=76 3+25=28 5+15=20$$

Ko te otinga te takirua ka hoatu i te tapeke 20.

$$a=5 b=15$$

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

$$\left(x+5\right)\left(x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x+5=0 me te x+15=0.

$$x=-5 x=-15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+75. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=20 ab=1\times 75=75$$

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 75.

$$1,75 3,25 5,15$$

Tātaihia te tapeke mō ia takirua.

$$1+75=76 3+25=28 5+15=20$$

Ko te otinga te takirua ka hoatu i te tapeke 20.

$$a=5 b=15$$

Tuhia anō te x^{2}+20x+75 hei \left(x^{2}+5x\right)+\left(15x+75\right).

$$\left(x^{2}+5x\right)+\left(15x+75\right)$$

Tauwehea te x i te tuatahi me te 15 i te rōpū tuarua.

$$x\left(x+5\right)+15\left(x+5\right)$$

Whakatauwehea atu te kīanga pātahi x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+5\right)\left(x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x+5=0 me te x+15=0.

$$x=-5 x=-15$$

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.

$$x^{2}+20x+75=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 20 mō b, me 75 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 75}}{2}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\times 75}}{2}$$

Whakareatia -4 ki te 75.

$$x=\frac{-20±\sqrt{400-300}}{2}$$

Tāpiri 400 ki te -300.

$$x=\frac{-20±\sqrt{100}}{2}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-20±10}{2}$$

Nā, me whakaoti te whārite x=\frac{-20±10}{2} ina he tāpiri te ±. Tāpiri -20 ki te 10.

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

Whakawehe -10 ki te 2.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-20±10}{2} ina he tango te ±. Tango 10 mai i -20.

$$x=-\frac{30}{2}$$

Whakawehe -30 ki te 2.

$$x=-15$$

Kua oti te whārite te whakatau.

$$x=-5 x=-15$$

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.

$$x^{2}+20x+75=0$$

Me tango 75 mai i ngā taha e rua o te whārite.

$$x^{2}+20x+75-75=-75$$

Mā te tango i te 75 i a ia ake anō ka toe ko te 0.

$$x^{2}+20x=-75$$

Whakawehea te 20, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 10. Nā, tāpiria te pūrua o te 10 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+20x+10^{2}=-75+10^{2}$$

Pūrua 10.

$$x^{2}+20x+100=-75+100$$

Tāpiri -75 ki te 100.

$$x^{2}+20x+100=25$$

Tauwehea x^{2}+20x+100. 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}.

$$\left(x+10\right)^{2}=25$$

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

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x+10=5 x+10=-5$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$x=-5 x=-15$$