Whakaoti i te n^2+5n=500

$$n ^ { 2 } + 5 n = 500$$

Whakaoti mō n

$$n=-25$$

$$n=20$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 500 mai i ngā taha e rua.

$$n^{2}+5n-500=0$$

Hei whakaoti i te whārite, whakatauwehea te n^{2}+5n-500 mā te whakamahi i te tātai n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=-500$$

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 -500.

$$-1,500 -2,250 -4,125 -5,100 -10,50 -20,25$$

Tātaihia te tapeke mō ia takirua.

$$-1+500=499 -2+250=248 -4+125=121 -5+100=95 -10+50=40 -20+25=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-20 b=25$$

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

$$\left(n-20\right)\left(n+25\right)$$

Hei kimi otinga whārite, me whakaoti te n-20=0 me te n+25=0.

$$n=20 n=-25$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 500 mai i ngā taha e rua.

$$n^{2}+5n-500=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei n^{2}+an+bn-500. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=1\left(-500\right)=-500$$

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 -500.

$$-1,500 -2,250 -4,125 -5,100 -10,50 -20,25$$

Tātaihia te tapeke mō ia takirua.

$$-1+500=499 -2+250=248 -4+125=121 -5+100=95 -10+50=40 -20+25=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-20 b=25$$

Tuhia anō te n^{2}+5n-500 hei \left(n^{2}-20n\right)+\left(25n-500\right).

$$\left(n^{2}-20n\right)+\left(25n-500\right)$$

Tauwehea te n i te tuatahi me te 25 i te rōpū tuarua.

$$n\left(n-20\right)+25\left(n-20\right)$$

Whakatauwehea atu te kīanga pātahi n-20 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-20\right)\left(n+25\right)$$

Hei kimi otinga whārite, me whakaoti te n-20=0 me te n+25=0.

$$n=20 n=-25$$

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.

$$n^{2}+5n=500$$

Me tango 500 mai i ngā taha e rua o te whārite.

$$n^{2}+5n-500=500-500$$

Mā te tango i te 500 i a ia ake anō ka toe ko te 0.

$$n^{2}+5n-500=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 5 mō b, me -500 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-5±\sqrt{5^{2}-4\left(-500\right)}}{2}$$

Pūrua 5.

$$n=\frac{-5±\sqrt{25-4\left(-500\right)}}{2}$$

Whakareatia -4 ki te -500.

$$n=\frac{-5±\sqrt{25+2000}}{2}$$

Tāpiri 25 ki te 2000.

$$n=\frac{-5±\sqrt{2025}}{2}$$

Tuhia te pūtakerua o te 2025.

$$n=\frac{-5±45}{2}$$

Nā, me whakaoti te whārite n=\frac{-5±45}{2} ina he tāpiri te ±. Tāpiri -5 ki te 45.

$$n=\frac{40}{2}$$

Whakawehe 40 ki te 2.

$$n=20$$

Nā, me whakaoti te whārite n=\frac{-5±45}{2} ina he tango te ±. Tango 45 mai i -5.

$$n=-\frac{50}{2}$$

Whakawehe -50 ki te 2.

$$n=-25$$

Kua oti te whārite te whakatau.

$$n=20 n=-25$$

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.

$$n^{2}+5n=500$$

Whakawehea te 5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{2}. Nā, tāpiria te pūrua o te \frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+5n+\left(\frac{5}{2}\right)^{2}=500+\left(\frac{5}{2}\right)^{2}$$

Pūruatia \frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+5n+\frac{25}{4}=500+\frac{25}{4}$$

Tāpiri 500 ki te \frac{25}{4}.

$$n^{2}+5n+\frac{25}{4}=\frac{2025}{4}$$

Tauwehea n^{2}+5n+\frac{25}{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}.

$$\left(n+\frac{5}{2}\right)^{2}=\frac{2025}{4}$$

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

$$\sqrt{\left(n+\frac{5}{2}\right)^{2}}=\sqrt{\frac{2025}{4}}$$

Whakarūnātia.

$$n+\frac{5}{2}=\frac{45}{2} n+\frac{5}{2}=-\frac{45}{2}$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$n=20 n=-25$$