Whakaoti i te 7n^2-121n-128-3728geq0

$$7 n ^ { 2 } - 121 n - 128 - 3728 \geq 0$$

Whakaoti mō n

$$n\in (-\infty,\frac{121-\sqrt{122609}}{14}]\cup [\frac{\sqrt{122609}+121}{14},\infty)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3728 i te -128, ka -3856.

$$7n^{2}-121n-3856\geq 0$$

Kia whakaotia te koreōrite, me tauwehe te taha mauī. Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$7n^{2}-121n-3856=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 7 mō te a, te -121 mō te b, me te -3856 mō te c i te ture pūrua.

$$n=\frac{-\left(-121\right)±\sqrt{\left(-121\right)^{2}-4\times 7\left(-3856\right)}}{2\times 7}$$

Mahia ngā tātaitai.

$$n=\frac{121±\sqrt{122609}}{14}$$

Whakaotia te whārite n=\frac{121±\sqrt{122609}}{14} ina he tōrunga te ±, ina he tōraro te ±.

$$n=\frac{\sqrt{122609}+121}{14} n=\frac{121-\sqrt{122609}}{14}$$

Tuhia anō te koreōrite mā te whakamahi i ngā otinga i whiwhi.

$$7\left(n-\frac{\sqrt{122609}+121}{14}\right)\left(n-\frac{121-\sqrt{122609}}{14}\right)\geq 0$$

Kia ≥0 te otinga, me ≤0 tahi, me ≥0 tahi rānei te n-\frac{\sqrt{122609}+121}{14} me te n-\frac{121-\sqrt{122609}}{14}. Whakaarohia te tauira ina he ≤0 tahi te n-\frac{\sqrt{122609}+121}{14} me te n-\frac{121-\sqrt{122609}}{14}.

$$n-\frac{\sqrt{122609}+121}{14}\leq 0 n-\frac{121-\sqrt{122609}}{14}\leq 0$$

Te otinga e whakaea i ngā koreōrite e rua ko n\leq \frac{121-\sqrt{122609}}{14}.

$$n\leq \frac{121-\sqrt{122609}}{14}$$

Whakaarohia te tauira ina he ≥0 tahi te n-\frac{\sqrt{122609}+121}{14} me te n-\frac{121-\sqrt{122609}}{14}.

$$n-\frac{121-\sqrt{122609}}{14}\geq 0 n-\frac{\sqrt{122609}+121}{14}\geq 0$$

Te otinga e whakaea i ngā koreōrite e rua ko n\geq \frac{\sqrt{122609}+121}{14}.

$$n\geq \frac{\sqrt{122609}+121}{14}$$

Ko te otinga whakamutunga ko te whakakotahi i ngā otinga kua whiwhi.

$$n\leq \frac{121-\sqrt{122609}}{14}\text{; }n\geq \frac{\sqrt{122609}+121}{14}$$