Whakaoti i te x^2+4x+3=5040

$${ x }^{ 2 } +4x+3=5040$$

Whakaoti mō x

$$x=-73$$

$$x=69$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 5040 mai i ngā taha e rua.

$$x^{2}+4x+3-5040=0$$

Tangohia te 5040 i te 3, ka -5037.

$$x^{2}+4x-5037=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+4x-5037 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.

$$a+b=4 ab=-5037$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -5037.

$$-1,5037 -3,1679 -23,219 -69,73$$

Tātaihia te tapeke mō ia takirua.

$$-1+5037=5036 -3+1679=1676 -23+219=196 -69+73=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-69 b=73$$

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

$$\left(x-69\right)\left(x+73\right)$$

Hei kimi otinga whārite, me whakaoti te x-69=0 me te x+73=0.

$$x=69 x=-73$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 5040 mai i ngā taha e rua.

$$x^{2}+4x+3-5040=0$$

Tangohia te 5040 i te 3, ka -5037.

$$x^{2}+4x-5037=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-5037. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=1\left(-5037\right)=-5037$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -5037.

$$-1,5037 -3,1679 -23,219 -69,73$$

Tātaihia te tapeke mō ia takirua.

$$-1+5037=5036 -3+1679=1676 -23+219=196 -69+73=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-69 b=73$$

Tuhia anō te x^{2}+4x-5037 hei \left(x^{2}-69x\right)+\left(73x-5037\right).

$$\left(x^{2}-69x\right)+\left(73x-5037\right)$$

Tauwehea te x i te tuatahi me te 73 i te rōpū tuarua.

$$x\left(x-69\right)+73\left(x-69\right)$$

Whakatauwehea atu te kīanga pātahi x-69 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-69\right)\left(x+73\right)$$

Hei kimi otinga whārite, me whakaoti te x-69=0 me te x+73=0.

$$x=69 x=-73$$

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.

$$x^{2}+4x+3=5040$$

Me tango 5040 mai i ngā taha e rua o te whārite.

$$x^{2}+4x+3-5040=5040-5040$$

Mā te tango i te 5040 i a ia ake anō ka toe ko te 0.

$$x^{2}+4x+3-5040=0$$

Tango 5040 mai i 3.

$$x^{2}+4x-5037=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 4 mō b, me -5037 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-5037\right)}}{2}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-5037\right)}}{2}$$

Whakareatia -4 ki te -5037.

$$x=\frac{-4±\sqrt{16+20148}}{2}$$

Tāpiri 16 ki te 20148.

$$x=\frac{-4±\sqrt{20164}}{2}$$

Tuhia te pūtakerua o te 20164.

$$x=\frac{-4±142}{2}$$

Nā, me whakaoti te whārite x=\frac{-4±142}{2} ina he tāpiri te ±. Tāpiri -4 ki te 142.

$$x=\frac{138}{2}$$

Whakawehe 138 ki te 2.

$$x=69$$

Nā, me whakaoti te whārite x=\frac{-4±142}{2} ina he tango te ±. Tango 142 mai i -4.

$$x=-\frac{146}{2}$$

Whakawehe -146 ki te 2.

$$x=-73$$

Kua oti te whārite te whakatau.

$$x=69 x=-73$$

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.

$$x^{2}+4x+3=5040$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x^{2}+4x+3-3=5040-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$x^{2}+4x=5040-3$$

Tango 3 mai i 5040.

$$x^{2}+4x=5037$$

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

$$x^{2}+4x+2^{2}=5037+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=5037+4$$

Tāpiri 5037 ki te 4.

$$x^{2}+4x+4=5041$$

Tauwehea x^{2}+4x+4. 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}.

$$\left(x+2\right)^{2}=5041$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{5041}$$

Whakarūnātia.

$$x+2=71 x+2=-71$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=69 x=-73$$