Whakaoti i te x^2-28x-480=0

$$x ^ { 2 } - 28 x - 480 = 0$$

Whakaoti mō x

$$x=-12$$

$$x=40$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-28x-480 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.

$$a+b=-28 ab=-480$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -480.

$$1,-480 2,-240 3,-160 4,-120 5,-96 6,-80 8,-60 10,-48 12,-40 15,-32 16,-30 20,-24$$

Tātaihia te tapeke mō ia takirua.

$$1-480=-479 2-240=-238 3-160=-157 4-120=-116 5-96=-91 6-80=-74 8-60=-52 10-48=-38 12-40=-28 15-32=-17 16-30=-14 20-24=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -28.

$$a=-40 b=12$$

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

$$\left(x-40\right)\left(x+12\right)$$

Hei kimi otinga whārite, me whakaoti te x-40=0 me te x+12=0.

$$x=40 x=-12$$

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-480. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-28 ab=1\left(-480\right)=-480$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -480.

$$1,-480 2,-240 3,-160 4,-120 5,-96 6,-80 8,-60 10,-48 12,-40 15,-32 16,-30 20,-24$$

Tātaihia te tapeke mō ia takirua.

$$1-480=-479 2-240=-238 3-160=-157 4-120=-116 5-96=-91 6-80=-74 8-60=-52 10-48=-38 12-40=-28 15-32=-17 16-30=-14 20-24=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -28.

$$a=-40 b=12$$

Tuhia anō te x^{2}-28x-480 hei \left(x^{2}-40x\right)+\left(12x-480\right).

$$\left(x^{2}-40x\right)+\left(12x-480\right)$$

Tauwehea te x i te tuatahi me te 12 i te rōpū tuarua.

$$x\left(x-40\right)+12\left(x-40\right)$$

Whakatauwehea atu te kīanga pātahi x-40 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-40\right)\left(x+12\right)$$

Hei kimi otinga whārite, me whakaoti te x-40=0 me te x+12=0.

$$x=40 x=-12$$

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.

$$x^{2}-28x-480=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -28 mō b, me -480 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\left(-480\right)}}{2}$$

Pūrua -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\left(-480\right)}}{2}$$

Whakareatia -4 ki te -480.

$$x=\frac{-\left(-28\right)±\sqrt{784+1920}}{2}$$

Tāpiri 784 ki te 1920.

$$x=\frac{-\left(-28\right)±\sqrt{2704}}{2}$$

Tuhia te pūtakerua o te 2704.

$$x=\frac{-\left(-28\right)±52}{2}$$

Ko te tauaro o -28 ko 28.

$$x=\frac{28±52}{2}$$

Nā, me whakaoti te whārite x=\frac{28±52}{2} ina he tāpiri te ±. Tāpiri 28 ki te 52.

$$x=\frac{80}{2}$$

Whakawehe 80 ki te 2.

$$x=40$$

Nā, me whakaoti te whārite x=\frac{28±52}{2} ina he tango te ±. Tango 52 mai i 28.

$$x=-\frac{24}{2}$$

Whakawehe -24 ki te 2.

$$x=-12$$

Kua oti te whārite te whakatau.

$$x=40 x=-12$$

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.

$$x^{2}-28x-480=0$$

Me tāpiri 480 ki ngā taha e rua o te whārite.

$$x^{2}-28x-480-\left(-480\right)=-\left(-480\right)$$

Mā te tango i te -480 i a ia ake anō ka toe ko te 0.

$$x^{2}-28x=-\left(-480\right)$$

Tango -480 mai i 0.

$$x^{2}-28x=480$$

Whakawehea te -28, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -14. Nā, tāpiria te pūrua o te -14 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-28x+\left(-14\right)^{2}=480+\left(-14\right)^{2}$$

Pūrua -14.

$$x^{2}-28x+196=480+196$$

Tāpiri 480 ki te 196.

$$x^{2}-28x+196=676$$

Tauwehea x^{2}-28x+196. 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}.

$$\left(x-14\right)^{2}=676$$

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

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{676}$$

Whakarūnātia.

$$x-14=26 x-14=-26$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$x=40 x=-12$$