Whakaoti i te a^2-68a+225leq0

$$a ^ { 2 } - 68 a + 225 \leq 0$$

Whakaoti mō a

$$a\in \begin{bmatrix}34-7\sqrt{19},7\sqrt{19}+34\end{bmatrix}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$a^{2}-68a+225=0$$

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 1 mō te a, te -68 mō te b, me te 225 mō te c i te ture pūrua.

$$a=\frac{-\left(-68\right)±\sqrt{\left(-68\right)^{2}-4\times 1\times 225}}{2}$$

Mahia ngā tātaitai.

$$a=\frac{68±14\sqrt{19}}{2}$$

Whakaotia te whārite a=\frac{68±14\sqrt{19}}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$a=7\sqrt{19}+34 a=34-7\sqrt{19}$$

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

$$\left(a-\left(7\sqrt{19}+34\right)\right)\left(a-\left(34-7\sqrt{19}\right)\right)\leq 0$$

Kia ≤0 te otinga, me ≥0 rawa tētahi uara o a-\left(7\sqrt{19}+34\right) me a-\left(34-7\sqrt{19}\right), me ≤0 anō te uara o tētahi. Whakaarohia te tauira ina ko a-\left(7\sqrt{19}+34\right)\geq 0 me a-\left(34-7\sqrt{19}\right)\leq 0.

$$a-\left(7\sqrt{19}+34\right)\geq 0 a-\left(34-7\sqrt{19}\right)\leq 0$$

He teka tēnei mō tētahi a ahakoa.

$$a\in \emptyset$$

Whakaarohia te tauira ina ko a-\left(7\sqrt{19}+34\right)\leq 0 me a-\left(34-7\sqrt{19}\right)\geq 0.

$$a-\left(34-7\sqrt{19}\right)\geq 0 a-\left(7\sqrt{19}+34\right)\leq 0$$

Te otinga e whakaea i ngā koreōrite e rua ko a\in \left[34-7\sqrt{19},7\sqrt{19}+34\right].

$$a\in \begin{bmatrix}34-7\sqrt{19},7\sqrt{19}+34\end{bmatrix}$$

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

$$a\in \begin{bmatrix}34-7\sqrt{19},7\sqrt{19}+34\end{bmatrix}$$